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In this work, an efficient method for constructing a complete integral of thegeodesic Hamilton-Jacobi equation on pseudo-Riemannian manifolds with simplytransitive groups of motions is suggested. The method is based on using aspecial transition to canonical coordinates on coadjoint orbits of the group ofmotion. As a non-trivial example, we consider the problem of constructing acomplete integral of the geodesic Hamilton-Jacobi equation in theMcLenaghan-Tariq-Tupper spacetime. An essential feature of this example is thatthe Hamilton-Jacobi equation is not separable in the correspondingconfiguration space.

Constructing a complete integral of the Hamilton-Jacobi equation on pseudo-Riemannian spaces with simply transitive groups of motions