  The higher-order superintegrability of the Tremblay-Turbiner-Winternitz
system (related to the harmonic oscillator) is studied on the two-dimensional
spherical and hiperbolic spaces, $S_\k^2$ ($\k>0$), and $H_{\k}^2$ ($\k<0$).
The curvature $\kappa$ is considered as a parameter and all the results are
formulated in explicit dependence of $\kappa$. The idea is that the additional
constant of motion can be factorized as the product of powers of two particular
rather simple complex functions (here denoted by $M_r$ and $N_\phi$). This
technique leads to a proof of the superintegrability of the
Tremblay-Turbiner-Winternitz system on $S_\k^2$ ($\k>0$) and $H_{\k}^2$
($\k<0$), and to the explicit expression of the constants of motion.
