All-degree polynomial first-integral rigidity on a Darboux locus of a Chazy-type family

Abstract

We study polynomial first integrals of the two-parameter third-order family $p'''+p p''+(b+1)(p')^2+(q p^2+e)p'=0,\qquad e\in\C^\times.$ After the polynomial change of variables \(y=p'\), \(w=p''+p p'\), the associated three-dimensional vector field contains a distinguished Darboux locus \(b=6q-2\), characterized within a natural cubic ansatz by an explicit Darboux polynomial with cofactor \(-p\). On this locus we classify polynomial first integrability in \emph{all} polynomial degrees: a nonconstant polynomial first integral exists if and only if $q\in\left\{\frac13,\frac12,1\right\}.$ The proof has three main ingredients. First, a unit codimension-one minor gives a cyclic Smith cokernel for every transverse homological operator and, in particular, injectivity of restriction to the equilibrium line. Second, on the Darboux locus an exact scalar chart adapted to \(F_D\) converts every polynomial first integral directly into a one-variable compatibility problem. For every \(v\ge1\) this yields a closed product obstruction for the reciprocal family \(a=1/v\); finiteness of the critical points of the restricted polynomial leaves only the three stated values and an arithmetic tail \(q=-1/J\). Finally, the latter is eliminated by an infinite-order automorphism of a smooth endpoint conic on the Darboux surface. The three surviving equations are precisely constant-accessory forms of Chazy IV, V and VI. In the normalized affine coordinates of our preceding work they are exactly the three intersections of the Darboux locus with the previously identified stable polynomial-integrability divisors. The new result is the all-degree exclusion of every other point on the Darboux locus. The full two-parameter all-degree classification away from that locus is not claimed here.