Meromorphic exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The classification of the meromorphic traveling wave solutions for this equation is presented. All possible exact solutions of the generalized Brethenton equation are given

Exact solutions of equations for the Burgers hierarchy

Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole - Hopf transformation

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

We consider a family of non-autonomous second-order differential equations, which generalizes the Li\'enard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold--Liouville integrability of Hamiltonian systems. We illustrate our results 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.Comment: To appear in Journal of Mathematical Analysis and Application