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

Geometry of Lax pairs: particle motion and Killing-Yano tensors