  The affine coherent states quantization is a promising integral quantization
of Hamiltonian systems when the phase space includes at least one conjugate
pair of variables which takes values from a half-plane. Such a situation is
common for gravitational systems which include singularities. The construction
of the quantization map includes a one-to-one mapping of the half-plane onto
the affine group. Particular cases of this mapping define specific
parametrizations of the group. Our aim is showing that different such
parametrizations lead to unitarily inequivalent quantum theories. Depending on
the Hamiltonian system under consideration, this dependence could potentially
be used constructively.
