  Using our anholonomic frame deformation method, we show how generic
off-diagonal cosmological solutions depending, in general, on all spacetime
coordinates and undergoing a phase of ultra-slow contraction can be constructed
in massive gravity. In this paper, there are found and studied new classes of
locally anisotropic and (in) homogeneous cosmological metrics with open and
closed spatial geometries. The late time acceleration is present due to
effective cosmological terms induced by nonlinear off-diagonal interactions and
graviton mass. The off-diagonal cosmological metrics and related St\" uckelberg
fields are constructed in explicit form up to nonholonomic frame transforms of
the Friedmann-Lama\^itre-Robertson-Walker (FLRW) coordinates. We show that the
solutions include matter, graviton mass and other effective sources modelling
nonlinear gravitational and matter fields interactions in modified and/or
massive gravity, with polarization of physical constants and deformations of
metrics, which may explain certain dark energy and dark matter effects. There
are stated and analyzed the conditions when such configurations mimic
interesting solutions in general relativity and modifications and recast the
general Painlev\'e--ullstrand and FLRW metrics. Finally, we elaborate on a
reconstruction procedure for a subclass of off-diagonal cosmological solutions
which describe cyclic and ekpyrotic universes, with an emphasis on open issues
and observable signatures.
