The inverse problem of a mixed Li\'enard type nonlinear oscillator   equation from symmetry perspective
  In this paper, we discuss the inverse problem for a mixed Li\'enard type nonlinear oscillator equation $\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0$, where $f(x),\,g(x)$ and $h(x)$ are arbitrary functions of $x$. Very recently, we have reported the Lie point symmetries of this equation. By exploiting the interconnection between Jacobi last multiplier, Lie point symmetries and Prelle-Singer procedure we construct a time independent integral for the case exhibiting maximal symmetry from which we identify the associated conservative non-standard Lagrangian and Hamiltonian functions. The classical dynamics of the nonlinear oscillator is also discussed and certain special properties including isochronous oscillations are brought out. 
Ajey K. Tiwari, S. N. Pandey, V. K. Chandrasekar, M. Senthilvelan and   M. Lakshmanan
nlin.SI