  We propose a method for integrating the right-invariant geodesic flows on Lie
groups based on the use of a special canonical transformation in the cotangent
bundle of the group. We also describe an original method of constructing exact
solutions for the Klein - Gordon equation on unimodular Lie groups. Finally, we
formulate a theorem which establishes a connection between the special
canonical transformation and irreducible representations of Lie group. This
connection allows us to consider the proposed methods of integrating for
classical and quantum equations in the framework of a unified approach.
