  Superintegrable systems in two- and three-dimensional spaces of constant
curvature have been extensively studied. From these, superintegrable systems in
conformally flat spaces can be constructed by Staeckel transform. In this paper
a method developed to establish the superintegrability of the
Tremblay-Turbiner-Winternitz system in two dimensions is extended to higher
dimensions and a superintegrable system on a non-conformally-flat
four-dimensional space is found. In doing so, curvature corrections to the
corresponding classical potential are found to be necessary. It is found that
some subalgebras of the symmetry algebra close polynomially.
