  A class of left-invariant second order reversible systems with functional
parameter is introduced which exhibits the phenomenon of robust integrability:
an open and dense subset of the phase space is filled with invariant tori
carrying quasi-periodic motions, and this behavior persists under perturbations
within the class.
  Real-analytic volume preserving systems are found in this class which have
positive Lyapunov exponents on an open subset, and the complement filled with
invariant tori.
