Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N \gamma_n(q_1 p_1 q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $\gamma_n$ arbitrary coefficients, $N\in \mathbb N$. For $N=2$, being both $\gamma_1,\gamma_2$ different from zero, this reduces to Zernike system. prove that $\mathcal H_N$ always provides a superintegrable system (for any value of $N$) obtaining corresponding constants motion explicitly, which turn out be higher-order in momenta. Such results not only applied Euclidean plane, but also sphere hyperbolic plane. In latter curved spaces, $ is expressed geodesic polar coordinates showing such new can regarded as superposition isotropic 1:1 (Higgs) oscillator with even-order anharmonic oscillators plus another momentum-dependent potentials. Furthermore, symmetry algebra determined studied, giving rise $(2N-1)$th-order polynomial algebra. As byproduct, interpreted family perturbations Finally, it shown (and so well) endowed Poisson $\mathfrak{sl}(2,\mathbb R)$-coalgebra would allow for further possible generalizations discussed.