Polynomial First Integrals of Second-Order ODEs: Exact Criteria and Minimal Degree

This archive accompanies the manuscript “Polynomial First Integrals of Second-Order ODEs: Exact Criteria and Minimal Degree”. It contains supplementary derivations and test examples, MATLAB code for constructing polynomial first integrals through degree three and examining their minimal degree, and precomputed results from a finite-basis scan of 729 polynomial Liénard equations. The files comprise Supplementary.pdf, MPFI_TDNE.m, MPFI_LienardScan.m, MPFI_singleODE_v4.m, and the scan records in MAT and CSV formats. The scan reports 27 linear, 78 quadratic, and 6 cubic verified integrals. The remaining 618 cases represent unsuccessful searches within the selected basis, not proofs of nonexistence. The MATLAB implementations require the Symbolic Math Toolbox.