All-Order Polynomial First-Integral Rigidity on the Toda-Compatible Slice of a Chazy-Type Family

We classify, without an a priori degree bound, the polynomial first integrals of the autonomous third-order family $p'''+p p''+k(p')^2+\left(\frac{p^2}{4}-\sigma\right)p'=0, (k,\sigma)\in\C^2.$ This is the constant-shift slice characterized by $2K'(p)=p$ in the more general equation $p'''+pp''+k(p')^2+K(p)p'=0$; we call it the Toda-compatible slice because its homological reconstruction admits a finite Jacobi/Toda formulation, although the proof given here is independent of that reconstruction. After a polynomial change of variables the system is a lower-weight deformation of a $(1,2,3)$-quasihomogeneous derivation. We prove that every irreducible factor of a homogeneous polynomial first integral is itself a first integral, and that restriction to the equilibrium line is injective. The classification is then reduced to a reduced invariant curve of a planar foliation on the weighted-projective chart $\PP(1,2,3)$. The two nontrivial projective singularities yield a finite arithmetic classification once their local separatrix geometry is combined with the tangent-cone resonance relation. The result is the exact ring classification $\ker \mathscr X_{k,\sigma}=$ $\begin{cases}\C[\mathcal H_3],&k=1,\C[\mathcal H_4],&k=2,\C[\mathcal H_6],&k=17/4,\C,&\text{otherwise},\end{cases}$ for every $\sigma\in\C$, where the primitive generators have highest weighted degrees $3,4,6$, respectively, and are given explicitly. Thus a nonconstant polynomial first integral exists if and only if $k\in\{1,2,17/4\}$. Classical Chazy IV, V and VI also carry quasihomogeneous first integrals of degrees $3,4,6$, but the natural-rescaling invariant pair of the present homogeneous slice does not occur in Chazy's homogeneous list. To the best of our knowledge, the present all-degree ring classification on the slice $2K'=p$ has not previously been stated.