On the geometric and analytical properties of the anharmonic oscillator

Here we consider the anharmonic oscillator that is a dynamical system given by $y_{xx}+\delta y^{n}=0$. We demonstrate that to this equation corresponds a new example of a superintegrable two-dimensional metric with a linear and a transcendental first integrals. Moreover, we show that for particular values of $n$ the transcendental first integral degenerates into a polynomial one, which provides an example of a superintegrable metric with additional polynomial first integral of an arbitrary even degree. We also discuss a general procedure of how to construct a superintegrable metric with one linear first integral from an autonomous nonlinear oscillator that is cubic with respect to the first derivative. We classify all cubic oscillators that can be used in this construction. Furthermore, we study the Li\'enard equations that are equivalent to the anharmonic oscillator with respect to the point transformations. We show that there are nontrivial examples of the Li\'enard equations that belong to this equivalence class, like the generalized Duffing oscillator or the generalized Duffing--Van der Pol oscillator.