Integrability of polynomial vector fields and a dual problem

The paper’s Theorem 4.1 proves that T ∘ Dj = Aj ∘ T (j=1,2) with A1(V)=∑(pi−qj)Dij(V) and A2(V)=(∑(P̃* i+Q̃* j)Dij+qP*+pQ*)V, leading to A=T∘(D1+D2)∘T−1 and mapping first integrals to solutions of A(V)=0. The candidate’s solution reproduces this argument: it computes T∘D1 and T∘D2 on the graded basis singled out by T−1, obtains the same weights for D1, and rewrites the D2 part using Euler-type operators E1,E2 to match the paper’s A2 action on monomials. The only difference is notational (using E1,E2); on monomials this yields exactly the same action as the paper’s ∑(P̃* i+Q̃* j)Dij, and the conclusion ker D ↔ ker A follows immediately from conjugacy and T’s graded isomorphism. See the statements and proof steps in Theorem 4.1 with definitions of T (21), D1,D2 (9),(11),(17), and A1,A2 (22)–(26) in the paper .