  In this paper we consider the relation between symmetries and first integrals
of canonical Hamiltonian equations. Based on a newly established identity
(which is an analog of well known Noether's identity for Lagrangian approach),
this approach provides a simple and clear way to construct first integrals with
the help of symmetries of a Hamiltonian. The approach is illustrated by a
number of examples, including equations of the three-dimensional Kepler motion.
