  In this paper is considered a problem of defining natural star-products on
symplectic manifolds, admissible for quantization of classical Hamiltonian
systems. First, a construction of a star-product on a cotangent bundle to an
Euclidean configuration space is given with the use of a sequence of pair-wise
commuting vector fields. The connection with a covariant representation of such
a star-product is also presented. Then, an extension of the construction to
symplectic manifolds over flat and non-flat pseudo-Riemannian configuration
spaces is discussed. Finally, a coordinate free construction of related quantum
mechanical operators from Hilbert space over respective configuration space is
presented.
