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We propose a method for integrating the right-invariant geodesic flows on Liegroups based on the use of a special canonical transformation in the cotangentbundle of the group. We also describe an original method of constructing exactsolutions for the Klein - Gordon equation on unimodular Lie groups. Finally, weformulate a theorem which establishes a connection between the specialcanonical transformation and irreducible representations of Lie group. Thisconnection allows us to consider the proposed methods of integrating forclassical and quantum equations in the framework of a unified approach.

A method of integration for classical and quantum equations based on the  connection between canonical transformations and irreducible representations  of Lie groups