  The $N$-dimensional quantum Hamiltonian $ \hat{H} = -\frac{\hbar^2
{|\mathbf{q} } | }{2(\eta +| {\mathbf{q}} |)} {\mathbf{\nabla}}^2 -
\frac{k}{\eta + |{\mathbf{q}} |} $ is shown to be exactly solvable for any real
positive value of the parameter $\eta$. Algebraically, this Hamiltonian system
can be regarded as a new maximally superintegrable $\eta$-deformation of the
$N$-dimensional Kepler-Coulomb Hamiltonian while, from a geometric viewpoint,
this superintegrable Hamiltonian can be interpreted as a system on an
$N$-dimensional Riemannian space with nonconstant curvature. The eigenvalues
and eigenfunctions of the model are explicitly obtained, and the spectrum
presents a hydrogen-like shape for positive values of the deformation parameter
$\eta$ and of the coupling constant $k$.
