  We propose a new framework for constructing geometric and physical models on
nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry
and nonlinear connection structure. Explicit parametrizations of generic
off-diagonal metrics and linear and nonlinear connections define different
types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to
spinor fields and Dirac operators on nonholonomic manifolds motivates the
theory of Clifford algebroids defined as Clifford bundles, in general, enabled
with nonintegrable distributions defining the nonlinear connection. In this
work, we elaborate the algebroid spinor differential geometry and formulate the
(scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids.
The paper communicates new developments in geometrical formulation of physical
theories and this approach is grounded on a number of previous examples when
exact solutions with generic off-diagonal metrics and generalized symmetries in
modern gravity define nonholonomic spacetime manifolds with uncompactified
extra dimensions.
