  The problem of intertwined Hamiltonians in two dimensional curved spaces is
investigated. Explicit results are obtained for Euclidean plane,Minkowski
plane, Poincar{\' e} half plane ($AdS_2$), de Sitter Plane ($dS_2$), sphere,
and torus. It is shown that the intertwining operator is related to the Killing
vector fields and the isometry group of corresponding space. It is shown that
the intertwined potentials are closely connected to the integral curves of the
Killing vector fields. Two problems of considered as applications of the
formalism presented in the paper. The first one is the problem of Hamiltonians
with equispaced energy levels and the second one is the problem of Hamiltonians
whose spectrum are like the spectrum of a free particle.
