Integrability by quadratures in optimal control of a unicycle on hyperbolic plane

We consider the problem of integrability by quadratures of normal Hamiltonian system in sub-Riemannian problem on the groups of motions of hyperbolic plane or pseudo Euclidean plane which form the Lie group SH(2). The first step towards proof of integrability is to calculate the local representation of the Lie group SH(2) in canonical coordinates of second kind. Wei-Norman transformation is applied to obtain this local representation. The Wei-Norman representation shows that the left invariant control system defined on the Lie group SH(2) is equivalent to the motion of a unicycle on hyperbolic plane. Three integrals of motion satisfying the Liouville's integrability conditions are then calculated to prove that the normal Hamiltonian system is integrable by quadratures.