Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in C2

Publ. Mat. 45 (2001), 95–123 NONDEGENERATE LINEARIZABLE CENTRES OF COMPLEX PLANAR QUADRATIC AND SYMMETRIC CUBIC SYSTEMS IN C2 C. Christopher and C. Rousseau Abstract In this paper we consider complex differential systems in the plane, which are linearizable in the neighborhood of a nondegenerate centre. We find necessary and sufficient conditions for linearizability for the class of complex quadratic systems and for the class of complex cubic systems symmetric with respect to a centre. The sufficiency of these conditions is shown by exhibiting explicitly a linearizing change of coordinates, either of Darboux type or a generalization of it. 1. Introduction This paper originated from the interest of the two authors in isochronous centres ([CD], [MRT] and [MMR]). Several authors have made a systematic study of the isochronous centres inside certain classes of systems with centre ([CJ], [L], [P], [RT1], [De], [Sa1] and [Sa2]). The first four papers study isochronous centres inside quadratic systems and cubic systems symmetric with respect to its centre. The work of Devlin [De] deals with those centres for which there is an integrating factor (x2+y2)α. The papers by Sabatini give conditions for isochronicity in terms of commuting vector fields. These studies are made possible by the fact that the centre conditions are known in both these cases. Unfortunately, there are very few “natural” families in which the centre conditions are known, although a number of mechanisms producing strata of centres are well known. In [CD] the Kukles family of systems was studied: ˙x=−y ˙y=x+a1x2+a2xy +a3y2+a4x3+a5x2y+a6xy2+a7y3. (1.1) 2000 Mathematics Subject Classification. 34C, 58F. This work was supported by NSERC and FCAR. 96 C. Christopher, C. Rousseau An original feature of this work is that, although the centre conditions are not known, it is still possible to find necessary and sufficient conditions for an isochronous centre in this family. The paper [CD] brings to light other interesting questions, one of which is the starting point for our investigations here. In looking for the necessary conditions for an isochronous centre in the Kukles system, a new set of conditions was found which cannot be satisfied for real systems in the family. However if we allow the coefficients aito be complex, then the conditions can be satisfied, and the origin is again isochronous. The notion of isochronicity still makes sense in this context as long as we keep to real time. In [CD] it was shown that a singular point is a complex isochronous centre if and only if a whole punctured neighborhood of the point consists entirely of closed trajectories. This interesting result thus gives a geometric characterization of isochronicity. This is completely different from the real case, where any centre can be made isochronous by multiplying by a suitable positive function. In the complex case, multiplying by a non-constant function changes the geometry of the (real time) trajectories. Simultaneous to these investigations, there has been a growing interest in the complex centres of systems in C2, a centre being a non-degenerate singular point with zero trace and a local analytic first integral. Note that integrable saddles also come within this category. Complex centres in quadratic systems have been studied by several mathematicians: Dulac [Du], Liu and Li [LL] and more recently Farell [F]. Putting together the results of these works it becomes natural to ask what are the complex isochronous quadratic systems. This raises the need of a good definition of a complex isochronous centre. Using a real time for such a system is somewhat artificial, especially since it excludes the natural identification of centres and saddles when working over the complex numbers. Our starting point here, therefore, is to introduce the equivalent notion of linearizable centre. That is, a system whose centre can be reduced to its linear part by an analytic change of coordinates. In this paper we give necessary and sufficient conditions for linearizability of complex quadratic and symmetric cubic centres. The paper [MMR] introduces a new method to prove isochronicity via Darboux linearizing change of coordinates. A refinement of this method allows us to give explicit linearizing changes of coordinates for each of these conditions. One important contribution of these investigations to the study of isochronicity is that there seems to be a far wider range of linearizable Complex Linearizable Centres 97 centres if we allow the coefficients to be complex. This seems to be in contrast to the case of integrability, where the corresponding strata are very similar. However, if we consider linearizable saddles instead, then the complex and real cases are comparable. A simple reason for this situation is that, for real centres, the separatrices are conjugate and there can be no independence in how a real linearizing transformation can act on either one. In the complex case and in the case of a real saddle this independence is preserved. It seems, therefore, that a better understanding of isochronicity as a phenomena should be obtained from considering linearizable saddles in more detail. In all studies of this nature, there is a need to have “visible” phenomena to spur on investigation. In the case of a real isochronous centre, it is easy to see that no finite critical point can lie on the boundary of the period annulus attached to the centre; a result which has been used in several classification problems. If we consider linearizable saddles, then this phenomena only remains valid with complex time, and consequently loses its power. However, from our investigations we conjecture the following obstruction to linearizability, which we hope will provide an impetus to further investigation along these lines: No linearizable saddle can lie on a homoclinic loop or, more generally, on a monodromic graphic. 2. Generalities 2.1. Complex centres and linearizable centres. Definition 2.1. (1) A singular point of a complex analytic system ˙z=P(z,w) ˙w=Q(z,w) (2.1) in C2is a centre if the system has an analytic first integral in a neighborhood of the singular point. (2) A centre is nondegenerate if the system has a non-vanishing 1-jet at the singular point which is a Morse singular point of an analytic first integral. (3) A nondegenerate centre is linearizable if there exists an analytic change of coordinates in the neighborhood of the singular point, bringing the system to a linear system. Remark 2.2.A nondegenerate centre necessarily has two opposite eigenvalues. 98 C. Christopher, C. Rousseau From now on, we will always restrict ourselves to nondegenerate centres. Proposition 2.3. Let us suppose that the system (2.1) has a nondegenerate centre at the origin and is of the form ˙z=iz +p(z,w)=iz +o(|(z,w)|2) ˙w=−iw +q(z,w)=−iw +o(|(z,w)|2). (2.2) The following are equivalent: (1) The origin is linearizable; (2) There exists a neighborhood of the origin such that every trajectory inside that neighborhood (with time as a real parameter) is periodic (if this is so, then the period is a constant). (3) The origin is stable, in the sense stable in the future and stable in the past, for real time t. Proof: As pointed by the referee the equivalence of (1) and (3) is a consequence of a theorem of Cartan-CarathThetaodory (1932) (see for instance [M, Theorem 2.1]), stating that a system of the form (2.2) is linearizable if and only if the solution (z,w)=(0,0) is stable. As any trajectory of the linear system is periodic, which implies the same result for the linearizable system, we have that (1) implies (2). The last part follows from (2) implies (3). Note, that the linearizability condition here is stronger than integrability and is not equivalent to the linearizability of foliations defined by 1-forms. More details on this can be found in [CMR]. 2.2. Darboux linearization and its generalization. We start with some definitions. Definitions 2.4. (1) For equation (2.1) we define the differential operator Dacting on analytic functions F(z,w) defined in a neighborhood of (0,0) by D(F)=∂F ∂z P(z,w)+∂F ∂wQ(z,w).(2.3) (2) An invariant algebraic curve of the system (2.1) is a curve in C2 given by an equation F(z,w) = 0, with F(z,w)∈C[z,w] such that there exists K(z,w)∈Cn−1[z,w] satisfying DF(z,w)=F(z,w)K(z,w).(2.4) Complex Linearizable Centres 99 Here Cn−1[z,w] denotes the space of polynomials in zand wof degree ≤n−1 and complex coefficients. F(x, y) is also called a Darboux factor. (3) Any analytic function F(z,w) satisfying (2.4), for some K(z,w)∈ Cn−1[z,w], is a generalized Darboux factor. The polynomial K(F)= K(z,w) is called the cofactor of the Darboux factor. (4) A nonconstant function F(z,w) satisfying DF(z,w)≡0isafirst integral. (5) A Darboux function is a function Z(z,w) of the form Z= k  j=0 Fαj j,α j∈C,(2.5) with Fj∈C[z,w], j=0,... ,k. (6) Given a system (2.1) and the differential operator Ddefined by (2.3), a Darboux function (resp. generalized Darboux function) associated with the system (2.1) is a function Zof the form (2.5), with Fj= 0 invariant algebraic curves (resp. FjDarboux factors), j=0,... ,k. (7) A system is (generalized) Darboux integrable if it has a first integral which is a (generalized) Darboux function associated to it. Many of the strata of polynomial systems with a centre have a first integral which is a Darboux function, or a generalized Darboux function (cf. [C], [S1], [S2]). In practice, the Darboux factors that we are most interested in arise as limiting cases of Darboux functions, and can be expressed in the form eD/E, where Dand Eare polynomials [C]. Many examples of isochronous centres having Darboux first integrals are also given in [MRT]. The concept of Darboux linearizability is introduced in [MMR], which however is only concerned with real systems. In the real context a linearizing change of coordinates is hence given by a unique function Z=F(z,z). Here we must adapt the definitions to the fact that we are dealing with systems in C2. Definition 2.5. The system (2.2) is (generalized) Darboux linearizable if there exists a (generalized) Darboux change of coordinates (Z, W)=  k  j=0 Fαj j,   j=0 Gβj j ,(2.6) 100 C. Christopher, C. Rousseau regular at the origin, i.e. of the form (Z, W)=(z+o(|(z,w)|),w + o(|(z,w)|)) linearizing (2.2). Such a function Zis called a (generalized) Darboux linearizing change of coordinates. Remark 2.6.A (generalized) Darboux linearizable system is (generalized) Darboux integrable with first integral F(Z, W)=ZW. The following theorem characterizing Darboux linearizability is obtained exactly as the corresponding theorem in [MMR]: Theorem 2.7. (i) The system (2.2) is Darboux linearizable if and only if there exist invariant algebraic curves F0=0and G0=0 of the form F0(z,w)=z+o(|(z,w)|),G0(z,w)=w+o(|(z,w)|) and there exist invariant algebraic curves Fj=0,j∈J1,Gj=0, j∈J2where J1and J2are finite subsets of N(possibly void) such that Fj(0,0) =0,Gj(0,0) =0and K0+ j∈J1 αjKj=i L0+ j∈J2 βjLj=−i (2.7) where Kjis the cofactor of Fj,Ljis the cofactor Gjand αj,β j∈ C. The Darboux linearizing change of coordinates is then given by (Z, W)= F0 j∈J1 Fαj j,G 0 j∈J2 Gβj j .(2.8) (ii) The system (2.2) is generalized Darboux linearizable if and only if (2.7) is satisfied with the Kjand Ljcofactors of Darboux factors Fj and Gj. The linearizing change of coordinates is again given by (2.8). Proof: The proof goes exactly as in [MMR]. It follows from the fact that (Z, W)=(F(z,w),G(z,w)) is a linearizing change of coordinates if and only if the functions Fand Gare Darboux factors satisfying DF =i and DG =−i. It turned out, in several examples of Darboux integrable systems, that we could only find algebraic invariant curves or Darboux factors so that one equation of (2.7) is satisfied. In order to construct the linearizing change of coordinates in that case we prove the following lemma. Complex Linearizable Centres 101 Lemma 2.8. Suppose that the system (2.2) has a first integral H(z,w)= zw +o(|(z,w)|2), and that there exist (generalized) Darboux factors Fj such that the first equation of (2.7) is satisfied. Then a linearizing change of coordinate is given by (Z, W)= F0 j∈J1 Fαj j,H(z,w) F0j∈J1Fαj j .(2.9) Proof: Let us call F=F0j∈J1Fαj jand G=H(z,w) F0j∈J1 Fαj j . Then H=FG. Since His a first integral we have DH = 0. From DF =iwe then deduce that DG =−i. Remark 2.9.A common method to exhibit explicitly a first integral is to use the Darboux method (with invariant algebraic curves or Darboux factors). It may happen that all Darboux factors occuring in the expression of the first integral do not vanish at the origin. In that case the integral obtained H(z,w) does not vanish at the origin. The theorem has to be applied to the first integral H(z,w)=H(z,w)−H(0,0). Even if His a Darboux first integral it may occur that H, and hence Gis not a Darboux function. 2.3. A class of Darboux linearizable centres. Theorem 2.10. The system ˙z=iz +zn+awn ˙w=−iw +1 nzn−1w (2.10) has a linearizable centre at the origin. Proof: For each system, we have the invariant algebraic curves F1(z,w)= w= 0 and F2(z,w)=z−iawn/(n+ 1) = 0, with cofactors −i+ zn−1/n and i+zn−1respectively. A third invariant curve with cofactor K3(z,w)=(n−1)zn−1is given by F3(z,w)=1+h(z,wn), where h(z,W)=n−1 j=0 ajzjWn−1−j, with an−1=−i, aj=−i(j+1)a n(n−1−j)−jaj+1 j=0,...,n−2.(2.11) The Darboux linearizing change of coordinates is then given by: (Z, W)=F2(z,w) (F3(z,w))1/(n−1) ,w (F3(z,w))1/(n(n−1)) . (2.12) 102 C. Christopher, C. Rousseau 3. Linearizable complex quadratic systems Theorem 3.1. We consider a quadratic system in C2: ˙z=iz +c20z2+c11zw +c02w2 ˙w=−iw +d20z2+d11zw +d02w2. (3.1) The system has a linearizable centre if and only if one of the following conditions is satisfied Ic11 =d20 =d11 =0(3.2) II c11 =c02 =d11 =0(3.3) III c02 =d20 =0,c 20 −d11 =d02 −c11 =0(3.4) IV 7c11 −6d02 =7d11 −6c20 =7d2 11 −12d20d02 (3.5) =14d20c02 −3d11d02 =49d11c02 −18d2 02 =0 V2c20 −5d11 =2d02 −5c11 =15d2 11 +4d20d02 (3.6) =6d2 02 +25c02d11 =10d20c02 −9d11d02 =0 VI c11 =c02 =d02 =0(3.7) VII c20 =d20 =d11 =0(3.8) VIII c11 =d20 =d02 =c20 −2d11 =0(3.9) IX c20 =c02 =d11 =d02 −2c11 =0.(3.10) In all cases one separatrix of the origin is an algebraic curve. Proof: To prove the necessity of the conditions, we bring the system (3.1) to normal form ˙ Z=iZ + j≥1 cjZj+1Wj ˙ W=−iW + j≥1 djZjWj+1 (3.11) up to terms of order 7 under a change of coordinates (z,w)=(Z+ o(|Z, W|),W +o(|Z, W|)). If the system is linearizable then we must have a1=a2=a3=b1=b2=b3= 0. The computations of ajand bj, for j=1,2,3 were performed in Maple and Reduce and a factorised Gr¨obner basis produced. The results were checked carefully between both packages, and yield the conditions I–IX above. The sufficiency of the conditions is given below: for each case (3.2)– (3.10) we give a linearizing change of coordinates. All Darboux factors are noted Fiand their respective cofactors Ki. Complex Linearizable Centres 103 (1) c11 =d20 =d11 = 0. Let us first suppose that c20,d 02 =0. We then scale c20 =d02 = 1. (The case c02 = 0 corresponds to the Loud system (S1) in the notation of [MRT].) The system has four invariant lines: F1(z,w)=wK 1(z,w)=−i+w F2(z,w)=1+iw K2(z,w)=w F3,4=1−iz +B3,4wK 3,4(z,w)=z−iB3,4w, (3.12) where B3,4are the roots of B2−iB −c02 = 0. The first integral is given by H(z,w)=1−iz +B3w 1−iz +B4w(1 + iw)i(B3−B4).(3.13) Choosing H(z,w)=i1−H(z,w) B3−B4=zw +o(|z,w|2), the linearizing change of coordinates is given by (Z, W)=H(z,w)(1 + iw) w,w 1+iw.(3.14) We now consider the case d02 = 0 and c20 = 1 (after scaling). Darboux factors and cofactors are given by: F1(z,w)=wK 1(z,w)=−i F2,3(z,w)=1−iz ±wK 2,3(z,w)=z∓iw F4(z,w)=ewK4(z,w)=−iw. (3.15) This yields a first integral H(z,w)=e−2w1−iz +w 1−iz −w.(3.16) We let H(z,w)= 1 2i(H(z,w)−1) = zw+o(|z,w|2) yielding the linearizing change of coordinates (Z, W)=H(z,w) w,w.(3.17) The case c20 =d02 = 0 is contained in (7) below. (2) c11 =c02 =d11 = 0 is dual of (1) under (z,w,t)→ (w,z,−t). (3) c02 =d20 =0,c 20 −d11 =d02 −c11 = 0. In the case c20d02 =0 we can scale c20 =d02 = 1, yielding the Loud system (S2) in the notation 110 C. Christopher, C. Rousseau Conversely suppose that the saddle point of the system is not integrable. Modulo an analytic change of coordinates and division by a locally non-vanishing function we can bring the system to a normal form ˙x=x, ˙y=−y+Axkyk+1 +o(|(x, y)|2k+1)(5.2) with A= 0. We will see that this is an obstruction to find an involution. Indeed it can be argued that such an involution Thas a linear part of the form T1(x, y)=(by, x/b), with b= 0. We look for the involution as a power series T(x, y)=(X, Y )=(by +∞ r=2 hr(x, y), x/b+∞ r=2 kr(x, y)), where hr(x, y) and kr(x, y) are homogeneous polynomials in xand yof degree r. From the hypothesis we must have the relation ˙ X=−X. However the term in xkyk+1 of degre 2k+ 1 in the expression ˙ X+Xis always of the form bA = 0, which contradicts the existence of such a T. Remarks 5.2.(1) In fact the hypothesis of the theorem can be relaxed. For example, by considering the order (2k+1)-terms of equation ˙ Y as well as ˙ Xwe can assume only that the linear parts of T(in the expansion about the critical point) are involutive and that the transformed system is equal to the original system multiplied by some negative function. Details are left to the reader. (2) It is also possible to prove the converse part of the theorem directly, without using normal forms. We sketch the idea below. If T:(x, y)→ (T1(x, y),T 2(x, y)), then we take new variables φ(x, y)=(X, Y )=(T1(x, y)−x, T1(x, y)+x). We want to show that in the coordinates (X,Y ) the involution becomes S(X,Y )= (−X,Y ), with S=φ◦T◦φ−1. If we call R(X,Y )=(−X,Y ) the symmetry with respect to the Y-axis, this amounts to showing that R◦φ◦T(x, y)=φ(x, y), which is a straightforword consequence of the fact that Tis an involution. We thus obtain a new system with a corresponding reversing transformation S, and so the system must be of the form ˙ X=−Y−P(X2,Y),˙ Y=−X+XQ(X2,Y).(5.3) Now such a system arises from the system ˙ W=−2Y−2P(W, Y ),˙ Y=−1+Q(W, Y ),(5.4) via the transformation W=X2. Since this later system is nonsingular at the origin, it has a first integral H(W, Y )=W− Y2+o(W)+o(Y2), which can be pulled back to a first integral K(X,Y )=X2−Y2+o(|(X,Y )|2) of the original system. Complex Linearizable Centres 111 Theorem 5.3. We consider a quadratic system in R2with a saddle point at the origin with opposite eigenvalues ˙x=x+c20x2+c11xy +c02y2 ˙y=−y+d20x2+d11xy +d02y2. (5.5) The system is linearizable at the origin if and only if one of the conditions (3.2)–(3.10) is satisfied. The phase portraits are given in Figures 1–6. Proof: The system (5.5) is obtained from (3.1) by means of the transformation (x, y, T)=(−iz, −iw, it). We have a real system whenever the cjk are real. As the conditions are invariant under (x, y)→ (ax, by), the families I, II, VI–IX can be reduced to one-dimensional families, while the cases III–V can be reduced to 0-dimensional families. However for practical reasons it is simpler for cases I, II, VI and VII to draw the bifurcation diagram on one fourth of a 2-sphere. For case I we suppose c2 20 +c2 02 +d2 02 = 1 and c20,d 02 ≥0. The bifurcation diagram appears in Figure 1. The case II can be deduced easily from it. d02 =0 c20 c02 Figure 1 112 C. Christopher, C. Rousseau The phase portraits of case III, IV, V appear in Figures 2, 3, and 4 respectively. Figure 2 Figure 3 Complex Linearizable Centres 113 Figure 4 For case VI we suppose c2 20+d2 20+d2 11 =1,c20,d 20 ≥0. The bifurcation diagram appears on Figure 5. Case VII can be easily deduced from it. c20 =0 d20 d11 d20 +d11 =0 d20 −d11 =0 Figure 5 114 C. Christopher, C. Rousseau For case VIII we can scale d2 11 +c2 02 =1,d11 ≥0. The bifurcation diagram appears in Figure 6. Case IX follows from it. c20 =0 c20 >0c20 <0 d11 =0d11 =0 Figure 6 A natural question for us was to see where these linearizable saddles lie inside the strata of integrable saddles. The integrability conditions for a quadratic system with a saddle at the origin first appear in the work of Dulac [Du] in a case by case procedure. They have been simplified in [LL] and then further more in [Z]. The phase portraits of integrable saddle points of quadratic systems are systematically studied in [DGS]. Theorem 5.4 ([LL] and [Z]).The strata of integrable saddles of (5.5) are given by (A)c11 =d11 =0, (B)d11 +2c20 =c11 +2d02 =0, (C)c20c11 −d11d02 =c3 11d20 −d3 11c02 =0, (D)2c11 −d02 =c20 −2d11 =c02d20 −d11c11 =0. (5.6) The stratum (A)consists of systems generically having three invariant lines, allowing a Darboux first integral (called Lotka-Volterra in [Z]and harmonic in [LL]). Complex Linearizable Centres 115 The stratum (B)consists of Hamiltonian systems. The stratum (C)consists of systems generically having an invariant line and an invariant conic allowing a Darboux first integral. It is called “symmetric” in [LL](for reasons of symmetry in the calculations of Lyapunov quantities) and reversible in [Z]. The stratum (D)consists of systems having a Malkin first integral generically constructed from an invariant conic and an invariant cubic. Remark 5.5.When c11d11 = 0 it is not possible to find a symmetry axis in the usual sense and we must introduce a generalization of reversibility, in the same way as exponential factors appear as the limits of Darboux integrals. The effect of approaching these limiting cases within the stratum (C) is that the eigenvectors of the reversing transformation coalesce and the transformation becomes singular. Thus, in order to resolve this difficulty, we are led to consider the system as a blow-up of a simpler system. For example the substratum (C1)of(C): ˙x=x+c20x2 ˙y=−y+d20x2+d11xy, (5.7) is a particular case of the following theorem which has been obtained by an algebraic method in [LL] (symmetries in the calculations of Lyapunov constants). Theorem 5.6. The following class of integrable saddles lies in the closure of the strata of reversible systems: ˙x=x+P(x, y)=x(1 + p(x, y)) ˙y=−y+Q(x, y), (5.8) where we have either P(x, y)=  i>j i+j≥2 cijxiyj,Q(x, y)=  i>j−2 i+j≥2 dijxiyj,(5.9) with cr+1,r +dr,r+1 =0, or the conjugate system under (x, y, t)→ (y,x,−t). 116 C. Christopher, C. Rousseau Proof: Taking X=xand Y=xy, we obtain the system ˙ X=X1+pX, Y X ˙ Y=YpX, Y X+XQX, Y X. (5.10) The conditions imply that a common factor of Xcan be removed to leave an analytic system with a non critical point at the origin. It therefore has a local first integral Φ(X,Y )=Y+o(|(X,Y )|) which can be pulled back to a local first integral of the original system, Φ(x, xy)=xy + o(|(x, y)|2). Remarks 5.7.(1) In the corresponding case for real centres, such systems do not arise except in the trivial case ˙x=−y1+αi(x2+y2)i ˙y=x1+αi(x2+y2)i. (5.11) (2) It would be interesting to see if the corresponding notion of a limit of rational reversible systems would also give some new integrability conditions. Proposition 5.8. The linearizable saddles described in Theorem 5.3 lie in the stratum (A)for the cases I–II, the stratum (C)for the cases III–VII, and the stratum (D)for the cases VIII–IX. In the latter case the invariant cubic is reducible, yielding a line through the origin and a conic. Theorem 5.9. We consider a cubic system in R2symmetric with respect to a saddle point at the origin with opposite eigenvalues ˙x=x+c30x3+c21x2y+c12xy2+c03y3 ˙y=−y+d30x3+d21x2y+d12xy2+d03y3. (5.12) The system is linearizable at the origin if and only if (4.2) and one of the conditions (4.3)–(4.9) is satisfied. The phase portraits appear in Figures 7–12 (only the generic cases). Proof: For case I we suppose c2 12 +c2 03 +d2 03 =1,c03 ≥0, which yields a half 2-sphere. The bifurcation diagram appears in Figure 7. (Case II is dual.) Complex Linearizable Centres 117 d03 c12 =d03 c12 =−d03 c12 c03 =0 Figure 7 For case III we suppose c2 30+c2 12 = 1. The bifurcation diagram appears in Figure 8. c12 c30 Figure 8 118 C. Christopher, C. Rousseau For case IV we suppose c2 30+d2 03 = 1. The bifurcation diagram appears in Figure 9. c30 d03 Figure 9 Case V is zero-dimensional. We have the conditions c30c12 <0, d30c03 <0, c30d21 >0, c12d03 >0 which yields, after scaling, the two cases ˙x=x+7x3−3xy2−4y3 ˙y=−y+4x3+3x2y−7y3 (5.13) and ˙x=x−7x3+3xy2+4y3 ˙y=−y−4x3−3x2y+7y3. (5.14) Their respective phase portraits appear in Figures 10 and 11. Figure 10 Complex Linearizable Centres 119 Figure 11 For case VI we suppose c2 03 +d2 21 = 1 and c03 ≥0. The bifurcation diagram appears in Figure 12. (Case VII is dual.) c03 d21 Figure 12 Theorem 5.10 ([LL]).The strata of integrable saddles are given by (A)d21 +3c30 =c21 +d12 =c12 +3d03 =0, (B)c21 =d12 =c30 −3d21 =3c12 −d03 =3c03d30 −4d21d03 =0, (C)c21 +d12 =c30c12 −d21d03 =c2 30c03 +d30d2 03 =c2 12d30 +c03d2 21 =0. (5.15) The stratum (A)consists of Hamiltonian systems. The stratum (B)consists of systems having a Malkin first integral constructed generically from an invariant quartic and an invariant sextic which give a rational first integral. The stratum (C)consists of reversible systems (called symmetric in [LL]), possibly in the generalized sense of Theorem 5.6 or in the sense below.