On gauge transforms of autonomous ordinary differential equations

The paper’s identification criterion (Proposition 4) states that an analytic nonautonomous field q(t,x)=c(t)+C(t)x+∑_{j≥2}q_j(t,x) is a gauge transform of some autonomous analytic f iff there exist a constant matrix B and an invertible A(t) with A(0)=I such that Ȧ=CA−AB, c(t)=A(t)c(0), and q_j(t,x)=A(t)q_j(0,A(t)^{-1}x) for all j≥2; this follows by comparing homogeneous terms in the equality q(t,x)=ȦA^{-1}x+Af(A^{-1}x), and conversely by defining f(x)=c(0)+Bx+∑_{j≥2}q_j(0,x) (Propositions 3–4) . The candidate solution reproduces exactly this necessity-and-sufficiency argument, including the normalization A(0)=I (as in Remark 5) and termwise comparison of constant, linear, and higher-order homogeneous parts, thereby matching the paper’s proof structure step-for-step . Analyticity and local Taylor expansions are assumed as in the paper’s Section 4 setup, which the candidate also invokes explicitly . Hence both are correct and essentially identical in method.