First and Second Integrals of Hopf–Langford-Type Systems

The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable.