  The Eisenhart geometric formalism, which transforms an Euclidean natural
Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional
degree of freedom, is applied to the four families of quadratically
superintegrable systems with multiple separability in the Euclidean plane.
Firstly, the separability and superintegrability of such four geodesic
Hamiltonians ${\cal T}_r$ ($r=a,b,c,d$) in a three-dimensional curved space are
studied and then these four systems are modified with the addition of a
potential ${\cal U}_r$ leading to ${\cal H}_r={\cal T}_r +{\cal U}_r$.
Secondly, we study the superintegrability of the four Hamiltonians
$\widetilde{{\cal H}}_r= {\cal H}_r/ \mu_r$, where $\mu_r$ is a certain
position-dependent mass, that enjoys the same separability as the original
system ${\cal H}_r$. All the Hamiltonians here studied describe superintegrable
systems on non-Euclidean three-dimensional manifolds with a broken spherically
symmetry.
