  We study Bertrand space-times (BSTs), which have been proposed as viable
models of space-times seeded by galactic dark matter, in modified theories of
gravity. We first critically examine the issue of galactic rotation curves in
General Relativity, and establish the usefulness of BSTs to fit experimental
data in this context. We then study BSTs in metric $f(R)$ gravity and in
Brans-Dicke theories. For the former, the nature of the Newtonian potential is
established, and we also compute the effective equation of state and show that
it can provide good fits to some recent experimental results. For the latter,
we calculate the Brans-Dicke scalar analytically in some limits and numerically
in general, and find interesting constraints on the parameters of the theory.
Our results provide evidence for the physical nature of Bertrand space-times in
modified theories of gravity.
