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}), qβ,pββR2, 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 qβ,pββRn is sketched.Comment: 9 pages, no figures; pasections reorganized, references updated, some
misprints correcte