  A geometric formulation of the Lax pair equation on a curved manifold is
studied using phase space formalism. The corresponding (covariantly conserved)
Lax tensor is defined and the method of generation of constants of motion from
it is discussed. It is shown that when the Hamilton equations of motion are
used, the conservation of the Lax tensor translates directly to the well known
Lax pair equation, with one matrix identified with components of the Lax tensor
and the other matrix constructed from the (metric) connection. A generalization
to Clifford objects is also discussed. Nontrivial examples of Lax tensors for
geodesic and charged particle motion are found in spacetimes admitting hidden
symmetry of Killing--Yano tensors.
