  In this article we consider nonholonomic deformations of disk solutions in
general relativity to generic off-diagonal metrics defining knew classes of
exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid
symmetries and local anisotropy and define certain generalizations of manifolds
with Killing and/ or Lie algebra symmetries. For Lie algebroids, there are
structures functions depending on variables on a base submanifold and it is
possible to work with singular structures defined by the 'anchor' map. This
results in a number of new physical implications comparing with the usual
manifolds possessing Lie algebra symmetries defined by structure constants. The
spacetimes investigated here have two physically distinct properties: First,
they can give rise to disk type configurations with angular/ time/ extra
dimension gravitational polarizations and running constants. Second, they
define static, stationary or moving disks in nontrivial solitonic backgrounds,
with possible warped factors, additional spinor and/or noncommutative
symmetries. Such metrics may have nontrivial limits to 4D gravity with
vanishing, or nonzero torsion. The work develops the results of Ref.
gr-qc/0005025 and emphasizes the solutions with Lie algebroid symmetries
following similar constructions for solutions with noncommutative symmetries
gr-qc/0307103.
