How to find all elliptic solutions of an ODE: new solution of the cubic-quintic complex Ginzburg–Landau equation

Given a nonlinear $N$-th order algebraic ordinary differential equation (ODE) which fails the Painlevé test, a major problem in physics is to find explicitly its general analytic solution, i.e. the largest $M$ -parameter particular solution without movable critical singularities, with $M$ strictly lower than $N$. We present here two results and one application.
The first result follows from Clunie's lemma of Nevanlinna theory: under two assumptions which happen to be true for most physically relevant nonintegrable ODE's, any meromorphic solution is necessarily elliptic.