Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations (Q2325910)

The author considers the family of equations \[y_{zz}+f(z,y)y_z+g(z,y)=0,\] which generalizes the Liénard equation. The main aim of this work is to study this equation as a Lax integrable system and obtain necessary and sufficient conditions for the equation to possess a first integral, which is quadratic with respect to the first derivative. The results are illustrated by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.