On the generalization of classical Zernike system

We generalize the results obtained recently (Nonlinearity \underline{36} (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian $H=\vec{p}{}^2+F(\vec{q}\cdot \vec{p})$, $\vec{q},\vec{p}\in {\mathbb{R}}^2$, for any analytic function $F$. The additional integral of motion is constructed explicitly and shown to reduce to a polynomial in canonical variables for polynomial $F$. The generalization to the case $\vec{q},\vec{p}\in {\mathbb{R}}^n$ is sketched.