On isochronous cases of the Cherkas system and Jacobi's last multiplier

We consider a large class of polynomial planar differential equations proposed by Cherkas (1976 Differensial'nye Uravneniya 12 201-6), and show that these systems admit a Lagrangian description via the Jacobi last multiplier (JLM). It is shown how the potential term can be mapped either to a linear harmonic oscillator potential or into an isotonic potential for specific values of the coefficients of the polynomials. This enables the identification of the specific cases of isochronous motion without making use of the computational procedure suggested by Hill et al (2007 Nonlinear Anal.: Theor. Methods Appl. 67 52-69), based on the Pleshkan algorithm. Finally, we obtain a Lagrangian description and perform a similar analysis for a cubic system to illustrate the applicability of this procedure based on Jacobi's last multiplier.