Real analytic vector fields with first integral and separatrices

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Metadata of the article that will be visualized in OnlineFirst ArticleTitle Real analytic vector fields with first integral and separatrices Article Sub-Title Article CopyRight The Royal Academy of Sciences, Madrid (This will be the copyright line in the final PDF) Journal Name Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Corresponding Author Family Name Mol Particle Given Name Rogério Suffix Division Departamento de Matemática Organization Universidade Federal de Minas Gerais Address Av. Antônio Carlos, 6627, CP 702, Belo Horizonte, 30123-970, Brazil Phone Fax Email [email protected] URL ORCID http://orcid.org/0000-0001-7799-4494 Author Family Name Sánchez Particle Given Name Fernando Sanz Suffix Division Departamento de Álgebra, Análisis Matemático, Geometría y Topología Organization Universidad de Valladolid Address Paseo de Belén 7, 47011, Valladolid, Spain Phone Fax Email [email protected] URL ORCID Schedule Received 12 May 2018 Revised Accepted 24 January 2019 Abstract We prove that a germ of analytic vector field at that possesses a non-constant analytic first integral has a real formal separatrix. We provide an example which shows that such a vector field does not necessarily have a real analytic separatrix. Keywords (separated by '-') Real analytic vector field - First integral - Formal and analytic separatrix - Reduction of singularities - Index of vector fields Mathematics Subject Classification (separated by '-') 32S65 - 37F75 - 34Cxx - 14P15 Footnote Information To our teacher Felipe Cano, with immense gratitude. Rogério Mol partially supported by Pronex-Faperj, CNPq-Universal, CAPES-Mathamsud.Fernando Sanz Sánchez partially supported by Ministerio de Educación y Cultura, Spain, process MTM2013-46337-C2-1- P and MTM2016-77642-C2-1-P, and by Programa Hispano-Brasileño de Cooperación Interuniversitaria, process PHB2010-0122-PC. uncorrected proof RACSAM https://doi.org/10.1007/s13398-019-00639-y ORIGINAL PAPER Real analytic vector fields with first integral and separatrices Rogério Mol1·Fernando Sanz Sánchez2 Received: 12 May 2018 / Accepted: 24 January 2019 © The Royal Academy of Sciences, Madrid 2019 Abstract1 We prove that a germ of analytic vector field at (R3,0)that possesses a non-constant analytic 1 2 first integral has a real formal separatrix. We provide an example which shows that such a3 vector field does not necessarily have a real analytic separatrix. 2 4 Keywords Real analytic vector field ·First integral ·Formal and analytic separatrix ·5 Reduction of singularities ·Index of vector fields6 Mathematics Subject Classification 32S65 ·37F75 ·34Cxx ·14P157 1 Introduction8 In this paper we prove the following result:9 Theorem 1 Let X be a germ of real analytic vector field at (R3,0)that has an analytic first10 integral. Then X has a real formal separatrix. The statement is optimal in the sense that such11 a vector field X does not necessarily have a real analytic separatrix.12 Speaking in general terms, let Xbe a germ of real analytic vector field at (Rn,0).A13 real analytic separatrix of Xis a germ of irreducible analytic curve Ŵat 0 ∈Rnwhich is14 invariant by X.Ifγ(t)=(γ1(t),...,γ n(t)) ∈(tR{t})n\{0}is a parametrization of Ŵ,the15 To our teacher Felipe Cano, with immense gratitude. Rogério Mol partially supported by Pronex-Faperj, CNPq-Universal, CAPES-Mathamsud. Fernando Sanz Sánchez partially supported by Ministerio de Educación y Cultura, Spain, process MTM2013-46337-C2-1-P and MTM2016-77642-C2-1-P, and by Programa Hispano-Brasileño de Cooperación Interuniversitaria, process PHB2010-0122-PC. BRogério Mol [email protected] Fernando Sanz Sánchez [email protected]a.es 1Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, CP 702, Belo Horizonte 30123-970, Brazil 2Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Universidad de Valladolid, Paseo de Belén 7, 47011 Valladolid, Spain 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof R. Mol , F. S. Sánchez invariance condition is equivalent to saying that there exists h(t)∈R{t}such that X(γ (t)) =16 h(t)dγ dt (t)for any t,whereh(t)≡ 0 if and only if Ŵis not contained in the singular locus17 Sing(X)={p;X(p)=0}of X. Replacing R{t}by R[[t]], we obtain the concept of real18 formal separatrix. On the other hand, considering the canonical complexification of Xto a19 holomorphic vector field at (Cn,0)and changing Rto C, we have the concepts of complex20 holomorphic separatrix and complex formal separatrix, seen as objects in (tC{t})n\{0}and21 (tC[[t]])n\{0}, respectively.22 We also recall that a first integral of Xis a germ of function f:(Rn,0)→Rsuch that23 df(X)=0. The expression “analytic first integral” in Theorem 1could be interpreted either24 as “holomorphic first integral” or “real analytic first integral”. In fact, if h:(C3,0)→Cis25 a non-constant holomorphic first integral of (the complexification of) X, then one can check26 that the real traces of Re(h)and Im(h)are real analytic first integrals of Xwith at least one27 of them non-constant.28 Notice that in Theorem 1we may assume without loss of generality that Xhas an29 isolated singularity at 0, otherwise there is at least a real analytic separatrix of Xcon-30 tained in Sing(X). On the contrary, we do not assume necessarily that the singular locus31 Sing(df)={p;df(p)=0}of the first integral fof Xis isolated. However, taking into32 account that Sing(df)is invariant by the vector field X, we may assume that it has no33 one-dimensional real components (see below in Sect. 4for details).34 Analytic or formal separatrices may of course be defined for holomorphic vector fields.35 They are algebraically manipulable invariant objects which play a central role in the study36 of the local dynamics of the vector field. Let us briefly review some avatars of the problem37 of existence of separatrices, related to the situation of real vector fields.38 Planar case, n =2. First, the Separatrix Theorem of Camacho and Sad [7] asserts that39 a planar vector field always has a complex holomorphic separatrix, although it may not40 have formal real separatrices: take, for instance, the standard vector field of center-type,41 X=−y∂ ∂x+x∂ ∂y.In this example, Xhas an analytic first integral, showing that Theorem 142 is not true for planar vector fields. On the other hand, there are examples of planar real vector43 fields with real formal separatrices, none of them convergent. An explicit example could be44 found in [29, Example 3.7(3)]. Below, in Sect. 5, we provide other examples used for the45 proof of the second part of Theorem 1.46 It is also known that an analytic vector field Xat 0 ∈R2with Poincaré index equal to47 zero has a real formal separatrix. Below, in Proposition 8, we provide a generalization of48 this result for vector fields defined in singular analytic surfaces, which is one of the main49 ingredients of the proof of Theorem 1.50 Three dimensional case, n =3. Camacho–Sad’s Theorem is no longer valid in this case:51 Gómez–MontandLuengo in [13]haveconstructedafamilyofvector fields in (C3,0)without52 complex separatrices. They state the result for analytic separatrices, although the same proof53 works in order to show that any vector field in that family is actually devoid of complex54 formal separatrices. An explicit member of that family with real coefficients could be found55 in [27, p. 333]. As a consequence of Theorem 1, vector fields in Gómez–Mont and Luengo’s56 family with real coefficients cannot have non-constant holomorphic first integrals.57 As for the planar case, there are examples of analytic vector fields at (R3,0)with formal58 real separatrices, none of them convergent (i.e. without real analytic separatrices). An explicit59 example can be found in [8, p. 3]. We construct in Sect. 5another example which has,60 moreover, a non-constant analytic first integral. It will prove the second part of Theorem 1,61 that is, that the conclusion “formal” in the statement cannot be improved to “analytic”.62 We should mention that, in a recent paper, D. Cerveau and A. Lins Neto proved that a63 germ of complex analytic vector field in (C3,0), with isolated singularity, that is tangent64 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof Real analytic vector fields with first integral and separatrices to a holomorphic foliation of codimension one always has a complex analytic separatrix65 [11, Proposition 3]. This result implies in particular that any vector field Xas in Theorem 166 actually has a complex analytic separatrix, inasmuch as Xis tangent to the foliation df =0,67 where fis a first integral. Such a complex separatrix may not be a real one (once more by68 our example in Sect. 5below).69 Higher dimension, n ≥4. Families of holomorphic vector fields at (Cn,0)without com-70 plex separatrices (neither convergent nor formal) are constructed in [21] for any dimension71 n≥4, generalizing the three dimensional construction carried out in [13]. Each one of these72 families contains an explicit example with real coefficients.73 Examples of real analytic vector fields without real formal separatrices having analytic74 first integral can be constructed in any dimension n≥4, showing that the phenomenon75 depicted in Theorem 1is exclusive for dimension three. When n=2pis even, we consider a76 multicenter vector field, written in coordinates (x1,y1,...,xp,yp)as Zn=X1+···+Xp,77 where Xj=−yj∂ ∂xj+xj∂ ∂yj. When n=2p+3 is odd, p≥1, we take coordinates78 (x1,y1,...,xp,yp,x,y,z)and set Zn=Z2p+W,whereZ2pis a multicenter vector field79 in the variables (x1,y1,...,xp,yp)and Wis one of the examples of three dimensional real80 vector fields in Gómez-Mont and Luengo’s family written in the variables (x,y,z). Notice81 that, in both cases, Znhas f(x1,y1)=x2 1+y2 1as a first integral.82 Finally, concerning real analytic separatrices in any dimension, it is worth mentioning83 Moussu’s paper [26], where it is proved that an analytic gradient vector field at (Rn,0)84 always has a real analytic separatrix. Below, we describe some arguments of that result,85 those which are used in our proof of Theorem 1(concretely, in Proposition 4).86 Let us sketch the proof of Theorem 1and the plan of the article. Let Xbe as in the87 hypothesis of the statement, having isolated singularity, and assume that the first integral f88 of Xis such that its singular locus Sing(df)does not have one-dimensional real components.89 Using a Brunella’s result [6] which guarantees that Xhas a non-trivial orbit accumulating90 to the origin, we may assume, moreover, that the special fiber Z=f−1(f(0)) of fis not91 reduced to the single point 0 ∈R3. Under these assumptions, we prove, in Sect. 2, a technical92 result(Proposition4)whichcanbeframed in the contextofrealversionsof Milnor’sFibration93 Theorem [24]. Roughly speaking, it asserts that, in any sufficiently small neighborhood of94 the origin, fhas regular fibers with connected components which are simply connected95 and which accumulate to a given two-dimensional component of the special fiber Z.Our96 proof of Proposition 4requires some avatars of known results in the theory of reduction of97 singularities of analytic functions. We recall them in the form needed for our purposes.98 In Sect. 3, we define, for any two-dimensional component Lof Z, the index IL(X)of the99 restriction X|L, a generalization to singular surfaces of the usual notion of Poincaré index of100 a planar vector field at a singular point. It is not really a new notion, it corresponds in one or101 another equivalent way to a particular case of standard definitions of the index of a vector field102 in a singular invariant variety (see [5] for more information). Pushing the restricted vector103 field X|Lto nearby fibers, using homotopic invariance of the index and the aforementioned104 result about simply connected fibers, we show that IL(X)is equal to zero for at least one105 component L.106 In Sect. 4, we conclude the proof of the first part of Theorem 1proving that, given a107 two-dimensional component Lof Z, either there exists a formal separatrix of Xinside Lor108 IL(X)= 0 (Proposition 8below). Incidentally, we use again the reduction of singularities as109 presented in Sect. 2for the proof of this result. As mentioned before, it generalizes a known110 result of planar vector fields to the situation of vector fields in singular surfaces. It is related to111 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof R. Mol , F. S. Sánchez Bendixson’s formula for the computation of the Poincaré index using hyperbolic and elliptic112 sectors of the vector field at the singularity.113 Finally, in Sect. 5, we provide an explicit example of a vector field Xwith isolated sin-114 gularity at 0 ∈R3which has an analytic first integral but which does not have any real115 analytic separatrix. The difficult part to check is that the formal real separatrix of such an116 example does really diverge. For that, we use the Martinet–Ramis moduli of planar holomor-117 phic foliations of saddle-node type [18,23] and the computation of the tangent of the moduli118 map in Elizarov’s work [12]. We thank Loïc Teyssier for his comments and decisive remarks119 concerning these arguments and techniques.120 2 About the fibers of a real analytic function121 The main result in this section is Proposition 4below, a result on the geometry of the fibers122 of a real analytic function in R3. We provide a proof adapted to our situation which employs123 the reduction of singularities of analytic functions. Some of the arguments are inspired on124 those of the paper [15] and also on a part of Roche’s work [30] concerning Real Clemens125 Structures.126 Our starting point is the following result (see Aroca et al. [2], Hironaka [16]orBierstone127 and Milman [3,4]).128 Theorem 2 Let f :(Rn,0)→(R,0)be a non-zero real analytic function. There exists a129 neighborhood U of 0∈Rnand a sequence of blow-ups with closed analytic non-singular130 centers131 π:Mm πm −→ Mm−1 πm−1 −→··· π2 −→ M1π1 −→ U(1)132 such that f ◦π:Mm→Ris everywhere locally of monomial type, i.e. it can be written133 locally as a monomial times a unit in analytic coordinates. Moreover, if Yj−1is the center134 of πjfor j =1,...,m, and we define recursively the total divisor E jat stage j by E j=135 π−1 j(Ej−1∪Yj−1)with E0=∅,thenYjhas normal crossing with E jand it is contained in136 the singular locus Sing(dfj)of f j=f◦π1◦···◦πj,for j≥0,where f 0=f.137 In particular, if Z=f−1(0)and ˜ Z=Z\Sing(df)is assumed to be non-empty (thus ˜ Z138 is a smooth analytic hypersurface), then πrestricts to an analytic isomorphism from π−1(˜ Z)139 to ˜ Z.140 For our purposes, we will use real blow-ups instead of the usual (projective) blow-ups πj 141 in Theorem 2. In order to define properly a real blow-up, we must consider the category of real142 analytic manifolds with boundary and corners; i.e. manifolds locally defined in coordinate143 charts (x1,...,xn)as quadrants {xi1≥0,xi2≥0,...,xir≥0}and so that the changes of144 coordinates are analytic isomorphisms preserving the quadrants. The point is that a real blow-145 up (also called a “polar blow-up”) produces a boundary in the blown-up manifold, namely the146 inverse image of the center by the blow-up (called exceptional divisor), which corresponds147 to the set of half-directions (instead of directions) in the normal bundle of the center as a148 submanifold of the ambient space. Subsequent real blow-ups produce new boundaries which149 intersect old boundaries along corners.150 Let us recall the main definitions here (see for instance the recent reference [22]for151 details). First, we define the real blow-up, with closed non-singular center Y, on a real152 manifold without boundary M.Letπ:M1→Mbe the usual blow-up of Mwith center Y153 and let τ:M+ 1→M1be the orientable double covering of M1. The composition π◦τis an154 analytic map which ramifies along the divisor E=π−1(Y). Then the real blow-up of Mwith155 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof Real analytic vector fields with first integral and separatrices center Yis the restriction σ:M′ 1→Mof π◦τto only one sheet, so that M′ 1is an analytic156 manifold with boundary ∂M′ 1=E. Next, more generally, if Mis a real analytic manifold157 with boundary and corners and Y⊂Mis a non-singular analytic submanifold having normal158 crossings with ∂M, we may consider first Mimmersed in a real analytic manifold  Mwith no159 boundaries or corners of the same dimension (the immersion is locally uniquely determined160 up to analytic isomorphisms), so that ∂Mbecomes a normal crossing divisor of  Mand such161 that Yis sent into a non-singular submanifold  Y⊂ Mwith normal crossings with ∂Minside162  M. The real blow-up σ:M′→Mwith center Y⊂Mis the restriction of the real blow-up163 σ: M′→ Mwith center  Yto M′=σ−1(M\Y).164 With this construction in mind, we adapt Theorem 2to obtain a version which uses real165 blow-ups and which will be more convenient for us. Although we can consider general166 statements, we will concentrate on three-dimensional analytic functions with some extra167 condition concerning its singular locus.168 Fix a germ f:(R3,0)→(R,0)of analytic function. Consider the prime decomposition169 f=fn1 1fn2 2···fnr r, where each fjis an irreducible germ of analytic function, and let170 h=Red(f)=f1f2···fr. Notice that Z=f−1(0)=h−1(0). Assume the following171 property, that we call Reduced Isolated Singularity:172 (RIS). The germ of analytic set Z=f−1(0)is not reduced to {0}and Sing(dh)⊂{0}.173 Note that the hypothesis (RIS) implies that, in some neighborhood of the origin, the174 set Z\{0}is a non-singular two-dimensional analytic submanifold and that the irreducible175 components f−1 j(0)of Z, as germs of analytic sets, only intersect at 0. (The converse of this176 result is not true: take f=Red(f)=y3−x6for which the special fiber Z={y−x2=0}177 is a non-singular surface at every point and the z-axis is contained in Sing(df).) To be more178 precise, let ε>0 be sufficiently small such that fis defined and analytic in a neighborhood179 of the closed ball V=B(0,ε), and such that Z∩Vcuts transversally the boundary of V.By180 the Conic Structure Theorem (see Milnor [24]orvdDries[32] for a more general statement),181 the set (Z\{0})∩Vhas finitely many connected components, denoted by L1,L2,...,Lr,182 where each Liis a non-singular analytic surface immersed in Vwhose closure in Vis183 homeomorphic to the cone at 0 over the link Ci=∂V∩Li(a curve homeomorphic to S1).184 The germs of the components Liat 0 are well defined and do not depend on ε. We will use the185 same notation Lifor both the components of (Z\{0})∩V(for any given sufficiently small186 ε) and their germs. They will be called local components of the special fiber Z=f−1(0).187 Proposition 3 Let f :(R3,0)→(R,0)be a germ of analytic function that satisfies the188 hypothesis (RIS). Then, if ε>0is sufficiently small and V =B(0,ε), there is a sequence of189 real blow-ups (independent of ε)190 σ:M′ m σm −→ M′ m−1 σm−1 −→··· σ2 −→ M′ 1 σ1 −→ V,(2)191 such that the composition f ◦σis everywhere locally of monomial type and such that, if L192 is a local component of Z =f−1(0),wehave:193 (i) σ−1(L)is diffeomorphic to the half-open cylinder [0,1)×S1, where the boundary194 {0}×S1corresponds to the link C =L∩∂V.195 (ii) The strict transform L′=σ−1(L)of L=L∪{0}is a real analytic submanifold of M′ m 196 with boundary and corners, homeomorphic to the closed cylinder [0,1]×S1.197 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof R. Mol , F. S. Sánchez (iii) Denoting ∂L′=C∞∪σ−1(C)the two connected components of the boundary of L′,198 we have that L′cuts transversally the total divisor E′ malong C∞, which is a piecewise199 smooth analytic curve homeomorphic to S1.200 (iv) The strict transforms of two different local components do not intersect.201 Moreover, σ1is the real blow-up with center Y ′ 0={0}and, if Y ′ j−1is the center of σjfor j =202 2,...,m, and we define recursively the total divisor E′jat stage j by E′j=σ−1 j(E′j−1∪Y′ j−1)203 with E′ 0=∅, then, for any j ≥1,Y′ j⊂E′jand E′jis homeomorphic to the sphere S2.204 Proof Let f=fn1 1fn2 2···fnr rbe the prime decomposition of fas a germ and put205 h=Red(f)=f1f2···fr.Letε>0 be sufficiently small such that fis defined in a206 neighborhood of a closed ball V=B(0,ε), and such that Zcuts transversally the boundary207 of V. Assume moreover that Vis contained in a neighborhood where Theorem 2applies208 to h, so that we obtain a sequence of blow-ups πas in (1) with centers Y0,Y1,...,Ym−1,209 such that h◦πis everywhere locally of monomial type. Therefore, the composition f◦πis210 also everywhere locally of monomial type. Define the sequence (2) recursively as follows:211 σ1:M′ 1→Vis the real blow-up of Vwith center Y′ 0=Y0,σ2:M′ 2→M′ 1the real blow-up212 with center Y′ 1=Y1∩M′ 1, and so on. Since Sing(dh)⊂{0}, by the hypothesis (RIS), and213 since the center Yj−1of πjis contained in the singular locus of hj−1=h◦σ1◦···◦σj−1,we214 have that Y′ 0={0}and that Y′ j⊂E′jfor j≥1. We deduce then that E′j∼ =S2by recurrence215 on j, using the definition of real blow-up.216 Property (i) is a consequence of the mentioned Conic Structure Theorem, together with the217 fact that σ:M′ m\E′ m→V\{0}is a diffeomorphism since each center Y′ jis contained in E′j 218 for j≥0. To prove properties (ii) and (iii) we use the conclusion that π−1(Z∩V)is a normal219 crossing divisor, so that L′is contained in one of its components, a non-singular analytic220 surface which cuts transversally the components of the total divisor E′ m. Finally, for property221 (iv), notice that if L′ 1,L′ 2are the strict transforms of two different local components L1,L2of222 Zand L′ 1∩L′ 2=∅, then necessarily L′ 1∩L′ 2⊂ E′ m(since L′ 1,L′ 2and any component of E′ m 223 are components of the normal crossing divisor σ−1(Z∩V)). Hence σ−1(L1)∩σ−1(L2)=∅224 and also L1∩L2=∅, which is impossible by the hypothesis (RIS). ⊓⊔225 Proposition 4 Let f :(R3,0)→(R,0)be a germ of analytic function satisfying the hypoth-226 esis (RIS). Then there is a local component L of the special fiber Z and a neighborhood base227 Bof 0∈R3such that each U ∈Bis compact and satisfies the following property: there228 exists a family {FU λ}λ∈(0,δ),whereF U λis a connected component of a non-singular fiber of229 f|U, such that FU λis homeomorphic to a closed disc and such that FU λ λ→0 −−−→(L∪{0})∩U230 in the Hausdorff topology.231 Proof We prove that any closed ball V=B(0,ε), with ε>0 sufficiently small for which232 Proposition 3holds, contains a neighborhood Uwith the required properties of the statement.233 WeusenotationofProposition3sothat,if L1,...,Lrarethelocalcomponentsofthesingular234 fiber Z=f−1(0)and L′jis the strict transform of Lj,thenL′jis homeomorphic to the235 cylinder [0,1]×S1and L′j∩E′ mis a curve homeomorphic to S1. Moreover, L′ i∩L′j∩E′ m=∅236 if i= j.Letj0be such that one of the connected components of E′ m\L′j0∩E′ m,sayD,237 contains no curve L′j∩E′ mfor j= j0.Then=L′j0∪Dis homeomorphic to a closed238 disc.239 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof Real analytic vector fields with first integral and separatrices Theorem 1 is a consequence of Proposition 5and of the following result:474 Proposition 8 For any j ∈{1,2,...,r}, either there is a formal real separatrix of X con-475 tained in L jor ILj(X)= 0.476 Proof Fix j∈{1,2,...,r}and put for simplicity L=Lj,C=Cjetc. Assume that477 εis sufficiently small so that Proposition 3holds for V=B(0,ε).Thatis,thereexistsa478 sequence of real blow-ups σ:M′→Vsuch that L′=σ−1(L)is a real analytic surface with479 boundary and corners, homeomorphic to a closed cylinder [0,1]×S1,suchthatσinduces a480 diffeomorphism between σ−1(L)and L. The boundary of L′consists of the two components481 C′=σ−1(C)(the transform of the link of Lby σ)andD′=L′∩E′,whereE′=σ−1(0)482 is the exceptional divisor of σ. While C′is a smooth analytic curve, D′is only piecewise483 smooth analytic. Denote by J⊂D′the set of corners of D′, i.e., the set of points where D′ 484 is not smooth. Consider in L′the orientation induced from that of Lby σ. Up to considering485 another surface diffeomorphic to L′, we may assume that L′is a submanifold with boundary486 and corners inside the euclidean plane R2, with the standard orientation.487 The transformed vector field X′=σ∗(X|L)in L′\D′defines a one-dimensional singular488 analytic foliation F′which can be extended analytically to D′as an oriented foliation (i.e.,489 at any point p∈D′, there is an analytic vector field X′pin a neighborhood Vpof pin L′,490 with isolated singularities, generating F′and such that X′pand X′are equally oriented in491 Vp\D′) whose set of singular points Sing(F′)is finite and contained in D′. Moreover, using492 Seidenberg’s Theorem on reduction of singularities [31],anduptoconsideringnewblow-493 ups on L′at points of D′, we can assume that any point of Sing(F′)is a simple singularity494 (that is, the eigenvalues λ,μ of the corresponding linear part are real and satisfy μ= 0495 and λ/μ /∈Q>0) and that any connected component of D′\Jis either invariant for F′or496 everywhere transversal to F′.497 Suppose that there is no formal real separatrix of Xinside L. Then, at any point p∈D′,498 the formal separatrices of F′at p(of a generator X′pof F′) are contained in D′. In particular,499 any connected component of D′\Jis invariant for F′. Also taking into account that a simple500 singularity of a two-dimensional real vector field, with real eigenvalues, has exactly two501 transversal formal separatrices (both real, non-singular and tangent to the corresponding502 eigendirections), we have necessarily that Sing(F′)=Jand that the only formal separatrices503 of F′at any p∈Jare the two components of D′through the point p(thus, they are analytic504 separatrices). Notice that, since D′is contained in the boundary of L′, there are exactly505 two connected components of D′\Jlocally at p∈J, each of them is part of one of the506 separatrices of F′at p. Each connected component ℓof D′\Jis a non-singular oriented leaf507 of F′, going from α(ℓ) to ω(ℓ), both points in J. A singular point p∈Jis either a sink,508 asource or a saddle, depending if ω(ℓ) =ω(ℓ′)=p,α(ℓ) =α(ℓ′)=por the remaining509 cases, respectively, where ℓ, ℓ′are the two components of D′\Jwhich accumulate to p.510 Sinks an sources are jointly called nodes.Anode connection is a union511 τ=ℓ1∪···∪ℓr 512 where the ℓjare connected components of D′\Jsatisfying α(τ) := α(ℓ1)is a source,513 ω(τ) := ω(ℓr)is a sink and ω(ℓj)=α(ℓj+1)for j=1,...,r−1 (which are saddle 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof R. Mol , F. S. Sánchez points). By construction, there is a continuum of trajectories of X′in L′\D′accumulating to514 τand having α(τ),ω(τ) as the αand ω-limit set, respectively. See the figure below:515 516 517 By the nature of real blow-ups, the cardinal of Jis even and, if J=∅, the number of518 connected components of D′\Jand the number of node connections are also even.519 To conclude the proposition, let us prove that in this situation we have IL(X)>0. The520 index IL(X)canbecalculatedasfollows.LetSbeaclosedsimple curvein L′\D′surrounding521 D′and homotopic to C′in L′\D′with the standard orientation and let φ:S1→Sbe an522 orientation preserving homeomorphism. Then IL(X)is equal to the degree of the map523 θ:S1→S1,p→ X′(φ(p)) X′(φ(p)).524 Suppose, moreover, that Sis a differentiable curve having only finitely many tangencies with525 X′and let iand ebe, respectively, the number of interior and exterior tangencies (i.e., at526 such a tangency point q, the orbit of X,devoidofq, stays locally at qin the interior or in527 the exterior of S, respectively). Then, from Poincaré (see also Pugh’s work [28]), we can528 calculate the degree of the map θabove as529 deg(θ) =1+i−e 2.(5)530 We will finish by constructing a differentiable curve Swith a positive (even) number of531 interior tangencies and no exterior tangencies with the vector field X′.532 Let τbe a node connection in D′and let γ1,γ 2be two trajectories of X′in L′\D′, both533 having αand ω-limit equal to α(τ) and ω(τ), respectively, and such that γ2is inside the534 circle τ∪γ1. Using the flow-box theorem, we can construct a differentiable arc of curve535 Sτconnecting two different points of γ1, lying inside τ∪γ1except for its extremities and536 everywhere transversal to X′except for a point aτwhere it touches γ2.Thisisdepictedin537 the figure:538 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof Real analytic vector fields with first integral and separatrices 539 540 We choose the extremities of Sτsufficiently near the corresponding singular points α(τ),541 ω(τ), so that, in sufficiently small neighborhoods of the the node singularities of D′, the arcs542 Sτcan be jointed in a smooth way by small arcs transversal to X′. Thus, we produce a simple543 closed curve Ssurrounding D′with the required properties: Sis everywhere transversal to544 X′except for the points aτ, which are in fact interior tangencies of Swith the vector field545 X′, and there are as many of them as the number of node connections (an even number). ⊓⊔546 Remark 9 It is worth noticing that formula (5) is closely related to Bendixson’s formula for547 the index of a planar analytic vector field Xat the origin in R2:548 I(X)=1+e−h 2 549 where eis the number of “elliptic” sectors and his the number of “hyperbolic” sectors of550 X′at the origin (see Andronov et al. [1]). In fact, in our situation, if we collapse L′into a551 neighborhood of 0 ∈R2sending D′to the origin, the push-forward of X′gives a vector552 field (which can be continuously extended to the origin) having as many elliptic sectors as553 the number of node connections in D′and no hyperbolic sectors. This is an alternative proof554 of the last part of Proposition 8, after the observation that Bendixson’s formula extends to555 continuous vector fields which have finitely many sectors of elliptic, hyperbolic or parabolic556 type.557 5 Examples558 In this section we prove the second part of Theorem 1, that is, we provide examples of559 analytic vector fields at 0 ∈R3having an analytic non-constant first integral but not having560 analytic separatrices. Our examples are obtained as a one-parameter unfolding of a two-561 dimensional vector field which has a unique formal real separatrix which is not convergent.562 In the introduction, we have already discussed the existence of planar vector fields with such563 a property, for instance, Risler’s example in [29]. We need to modify such example in order564 that its unfolding produces a three-dimensional vector field with isolated singularity.565 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof R. Mol , F. S. Sánchez Proposition 10 Let a =a(x)∈R{x}be a convergent series in one variable such that566 a(0)=a′(0)=0and consider the planar analytic vector field567 Ya=(y2+x4)∂ ∂x+−xy +x3a(x)+a(x) xy2∂ ∂y.(6)568 Then Yahas a unique real formal separatrix Ŵaat 0∈R2and, for a convenient (in fact569 generic) choice of the series a(x),Ŵais not convergent.570 Using this proposition, we construct our desired examples in R3as follows.571 Example 11 Given a(x)∈R{x}with a(0)=a′(0)=0, consider the vector field in R3,572 expressed in coordinates (x,y,z)as573 Xa=Ya+z2∂ ∂x,574 where Yais given in (6). The vector field Xais in fact a family of planar vector fields in the575 parameter z. In other words, the function f=zis an analytic first integral of Xa. Moreover,576 since the coefficient of ∂/∂xis y2+x4+z2, the origin is an isolated singularity of Xa. Hence,577 the real formal separatrices of Xaare those contained in the fiber z=0. More specifically,578 they are the separatrices of the restriction Xa|z=0=Ya. By Proposition 10,Xahas a unique579 real formal separatrix Ŵa, which is not convergent for a convenient choice of the series a(x).580 Proof of Proposition 10 If Ŵis a formal real separatrix of Yathen its tangent line corresponds581 to a root of the tangent cone of Yaat the origin, which is given by the equation y3+yx2=582 y(y2+x2)=0. Thus Ŵis tangent to ℓ=(y=0).Letπ1:M1→R2be the blow-up at the583 origin and let p1be the point in the exceptional divisor E1=π−1 1(0)corresponding to ℓ.584 The strict transform Ŵof Ŵby π1is a formal separatrix of the the strict transform Yaof Ya 585 at p1. A computation using usual coordinates (x,y1)=(x,y/x)of the blow-up π1shows586 that Yahas a saddle-node singularity at p1for which the divisor E1is the strong separatrix587 (tangent to the non-zero eigenvalue) and thus Ŵis the weak formal separatrix (tangent to the588 zero eigenvalue). This proves the uniqueness of Ŵ=Ŵa.589 Let us prove that Ŵais not convergent for some choice of the series a(x).Forthat,we590 consider the blow-up π2:M2→M1at the point p1and the point p2in the exceptional591 divisor E2=π−1 2(p1)corresponding to the tangent of Ŵaat p1. We put usual coordinates at592 p2of the form (x,y2)=(x,y1/x)=(x,y/x2)and compute the strict transform of Yaas593 Ya=x3(1+y2 2)∂ ∂x+−y2(1+2x2(1+y2 2)) +a(x)(1+y2 2)∂ ∂y2.594 Again Yahas a saddle-node singularity for which the divisor E2=(x=0)is the strong595 separatrix and the strict transform Ŵaof Ŵaby π2is the weak separatrix. To finish, let us show596 that Ŵais not convergent for a convenient choice of a. Let us assume that a(x)=α(2x2)for597 some α(z)∈zR{z}. After dividing Yaby 1 +y2 2, we consider the ramification z=2x2and598 rename w=y2, obtaining the saddle-node vector field599 ξα=z2∂ ∂z+−w(1+z)+w3 1+w2+α(z)∂ ∂w.(7)600 It suffices to prove the following:601 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof Real analytic vector fields with first integral and separatrices Assertion There is a choice of the series α(z)so that, for any δ>0sufficiently small, the602 weak formal separatrix of the saddle-node vector field ξδα is not convergent.603 We use the Martinet-Ramis moduli for analytic orbital classification of holomorphic foli-604 ations generated by saddle-node vector fields at the origin of C2(see [23]andalso[18]). In605 our particular case, any vector field ξαof the form (7) is formally orbitally equivalent to the606 vector field in normal form607 N=z2∂ ∂z+(−w(1+z)) ∂ ∂w.608 If we denote by Nthe class of vector fields formally orbitally equivalent to N, the moduli609 map associates to any η∈Nis a couple G(η) =(g(η), ψ(η)) where g(η) ∈Cand ψ(η) is610 a germ of a tangent to the identity biholomorphism at (C,0)in such a way that two vector611 fields η, η′are orbitally analytically equivalent if and only if G(η) =G(η′). On the other612 hand, if η∈Nthen the weak formal separatrix of ηis convergent if and only if the constant613 part g(η) of the moduli is equal to zero [23, Theorem III.4.4]. Moreover, if we have a family614 {ηλ}of vector fields in Ndepending analytically on λ∈Cmthen λ→ g(ηλ)is also analytic615 [18, Theorem 1, p. 33].616 In order to prove the assertion, put δ=ε3/2for ε∈R>0and write the vector field ξδα 617 under the change of variable w=√ε¯was618 ηε,α =z2∂ ∂z+−¯w(1+z)+ε¯w3+α(z)−ε2¯w5+ε3¯w7−···∂ ∂¯w.619 Hence g(ξε3/2α)=g(ηε,α)and it suffices to show that there exists a series α=α(z)so that620 d(g(ηε,α)) dε|ε=0= 0.(8)621 (Notice that this gives the assertion since the weak separatrix of ξ0=η0,α is ¯w=0and622 hence g(ξ0)=0). First, put623 ηε,α =N+ε( ¯w3+α(z)) ∂ ∂¯w=z2∂ ∂z+−¯w(1+z)+ε( ¯w3+α(z))∂ ∂¯w,624 so that (changing the notation w=¯w)wehaveηε,α =ηε,α +εYεwhere625 Yν=(−νw5+ν2w7−···)∂ ∂w.626 In other words, if we put ζε,ν,α =ηε,α +εYνthen we have ηε,α =ζε,ε,α. Notice that, for any627 series α,wehavethatg(ζε,ν,α)is analytic in (ε, ν),g(ζ0,ν,α)=0foranyνand ζε,0,α =ηε,α 628 for any ε. Hence we obtain629 d(g(ηε,α)) dε|ε=0=d(g(ηε,α)) dε|ε=0.630 Thus, to prove (8), it suffices to show that d(g(ηε,α )) dε|ε=0= 0 for some choice of α.631 The derivative of g(ηε,α)at ε=0 (considered as a component of the tangent of the moduli632 map G) can be computed explicitly from Elizarov’s paper [12] as follows. Make the change633 of variables z→−zand multiply by −1, getting the new expression for the family634 ηε,α =z2∂ ∂z+w(1−z)−ε(w3+α(−z))∂ ∂w.635 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof R. Mol , F. S. Sánchez To put it in Elizarov’s pattern, we have to divide it by 1 −zso that the family ¯ηε,α becomes636 the family vp,λ +εP∂wconsidered in equation [12, Eq. 1.8], where637 vp,λ =z2 1−z ∂ ∂z+w∂ ∂w (and hence p=1andλ=−1)638 and639 P(z,w)=−α(−z)+w3 1−z=−(α(−z)+w3)(1+z+z2+···).640 Choose α(z)such that α(0)=α′(0)=0 and write641 −α(−z)(1+z+z2+···)= k≥2 ckzk.642 This corresponds to f−1(z)in the expansion in power series in [12, Eq. 1.9]. The constant643 part gof the moduli map corresponds in our case to the component a0,−1in equation [12,644 Eq. 1.3] (that is, j=0andl=−1).645 From all these data, and computing the sequence mk(l)=mk(−1)in [12, Eq. 1.7] for646 the corresponding Borel transform, we conclude from Elizarov’s formula in [12, Theorem647 1] that648 d(g(ηε,α)) dε|ε=0=u∞  k=2 ck k Ŵ(k+2),649 where Ŵis the Euler’s Gamma function and uis some non-zero constant which does not650 depend on α(if we want to be precise, we can check that in fact u=−1). 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Soc., Lecture Notes Series,710 p. 248 (1998)711 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and712 institutional affiliations.713 123 Journal: 13398 Article No.: 0639 TYPESET DISK LE CP Disp.:2019/1/30 Pages: 19 Layout: Small Author Proof uncorrected proof Journal: 13398 Article: 639 Author Query Form Please ensure you fill out your response to the queries raised below and return this form along with your corrections Dear Author During the process of typesetting your article, the following queries have arisen. Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proof/online grid or in the ‘Author’s response’ area provided below Query Details required Author’s response 1. Please confirm if the author [Fernando Sanz Sánchez] initial is correctly identified. Amend if necessary. 2. 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