Darboux integrability of a cubic differential system with two parallel invariant straight lines

In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.

An algebraic criterion of the Darboux integrability of differential-difference equations and systems

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A close connection is found between integrals and characteristic Lie-Rinehart algebras of the system. It is proved that a system of equations is Darboux integrable if and only if its characteristic algebras in both directions are finite-dimensional.