  The problem of an origin of the Solary-Kochetov extra-phase contribution to
the naive semiclassical form of a generalized phase-space propagator is
addressed with the special reference to the su(2) spin case which is the most
important in applications. While the extra-phase correction to a flat
phase-space propagator can straightforwardly be shown to appear as a difference
between the principal and the Weyl symbols of a Hamiltonian in the
next-to-leading order expansion in the semiclassical parameter, the same
statement for the semiclassical spin coherent-state propagator holds provided
the Holstein-Primakoff representation of the su(2) algebra generators is
employed.
