  The classical quantization of the motion of a free particle and that of an
harmonic oscillator on a double cone are achieved by a quantization scheme
[M.C. Nucci, Theor. Math. Phys. 168 (2011) 994], that preserves the Noether
point symmetries of the underlying Lagrangian in order to construct the
Schroedinger equation. The result is different from that given in [K. Kowalski,
J.Rembielnski, Ann. Phys. 329 (2013) 146]. A comparison of the different
outcomes is provided.
