Nonlocal Linearization and Conformal Lax Representations for Non-Autonomous Coupled Second-Order Systems

We study non-autonomous coupled second-order systems x′′=Π1(t,x,y,x′,y′), y′′=Π2(t,x,y,x′,y′) and the question of when such a system is carried by a generalized Sundman transformation (GST) u=φ(t,x,y), v=ψ(t,x,y), ds=η(t,x,y)dt onto a linear coupled system with constant coefficients, including damping and restoring terms. We state the determining system for (φ,ψ,η) and an explicit constancy test for its linear and zeroth-order part. For the part of the transformed equations that is quadratic in the velocities, we prove a geometric characterization: it is realized by a GST if and only if a torsion-free affine connection, depending on the gauge function G=log|η|, is flat. This condition is necessary but not sufficient for linearizability of the complete system. For the linear target, we describe all quadratic invariants through a Lyapunov-type matrix equation and give the correct count of independent ones, with the assumptions needed. A rank-one matrix L=ZZTS built from the transformed phase-space vector Z satisfies a conformal Lax equation L˙=[L,W]+κηL; we prove the converse implication under an explicit nondegeneracy condition. All trace invariants of this matrix are powers of a single quadratic form, so the Lax formalism reorganizes, rather than enlarges, the set of invariants. For a damped cyclotron, we obtain two exponentially weighted first integrals, two exactly decaying quadratic quantities, and one further first integral free of exponential weight. The numerical checks are documented with convergence tables, and a script reproducing all figures is supplied.