  We show that in the framework of R^{2} gravity and in the linearized approach
it is possible to obtain spherically symmetric stationary states that can be
used as a model for galaxies. The model can also help to have a better
understanding on the theoretical basis of Einstein-Vlasov systems.
Specifically, we discuss, in the linearized R^{2} gravity, the solutions of a
Klein-Gordon equation for the spacetime curvature. Such solutions describe high
energy scalarons, a field that in the context of galactic dynamics can be
interpreted like the no-light-emitting galactic component. That is, these
particles can be figured out like wave-packets showing stationary solutions in
the Einstein-Vlasov system. As pertinent to the issue under analysis in this
paper, we present an analysis on the gravitational lensing phenomena within
this framework. Although the main goal of this paper is to give a potential
solution to the Dark Matter Problem within galaxies, we add a Section where we
show that an important property of the Bullet Cluster can in principle be
explained in the scenario introduced in this work. To the end, we discuss the
generic prospective to give rise to the Dark Matter component of most galaxies
within extended gravity.
