On the Jacobi last multipliers and Lagrangians for a family of Liénard-type equations

We study a family of the Liénard-type equations, which can be transformed via the generalized Sundman transformations into a particular case of Painlevé-Gambier equation XXVII. We show that this equation of the Painlevé-Gambier type admits an autonomous Lagrangian, Jacobi last multiplier and first integral. As a consequence, we obtain that the corresponding family of Liénard-type equations also admits a time-independent Lagrangian, Jacobi last multiplier and first integral. We also construct the general analytical and singular solutions for members of this family of Liénard-type equations by virtue of the generalized Sundman transformations. To demonstrate applications of our results we consider several examples of the Liénard-type equations, with a generalization of the modified Emden equation among them, and construct their autonomous Lagrangians, Jacobi last multipliers and first integral as well as their general analytical solutions.