Isochronicity Conditions and Lagrangian Formulations of the Hirota Type Oscillator Equations

We consider the continuous version of the nonlinear Hirota oscillator equation and by using the Jacobi last multiplier (JLM) show that two types of Lagrangians (or Hamiltonians) can be derived corresponding to different choices of the JLM. The existence of a mirror partner (or sister) equation is shown. Both equations belong to the Liénard-II class and the Hirota oscillator equations are shown to emerge from the requirement of isochronicity.