  Bosonic quantum conversion systems can be modeled by many-particle
single-mode Hamiltonians describing a conversion of $n$ molecules of type A
into $m$ molecules of type B and vice versa. These Hamiltonians are analyzed in
terms of generators of a polynomially deformed $su(2)$ algebra. In the
mean-field limit of large particle numbers, these systems become classical and
their Hamiltonian dynamics can again be described by polynomial deformations of
a Lie algebra, where quantum commutators are replaced by Poisson brackets. The
Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres
with cusp singularities depending on $m$ and $n$. It is demonstrated that the
many-particle eigenvalues can be recovered from the mean-field dynamics using a
WKB type quantization condition. The many-particle state densities can be
semiclassically approximated by the time-periods of periodic orbits, which show
characteristic steps and singularities related to the fixed points, whose
bifurcation properties are analyzed.
