  A non-Hermitian PT-symmetric version of the kicked top is introduced to study
the interplay of quantum chaos with balanced loss and gain. The classical
dynamics arising from the quantum dynamics of the angular momentum expectation
values are derived. The presence of PT-symmetry can lead to stable mixed
regular chaotic behaviour without sinks or sources for subcritical values of
the gain-loss parameter. This is an example of what is known in classical
dynamical systems as reversible dynamical systems. For large values of the
kicking strength a strange attractor is observed that also persists if
PT-symmetry is broken. The intensity dynamics of the classical map is found to
provide the main structure for the Husimi distributions of the subspaces of the
quantum system belonging to certain ranges of the imaginary parts of the
quasienergies. Classical structures are identified in the quantum dynamics.
Finally, the statistics of the eigenvalues of the quantum system are analysed
and it is shown that if most of the eigenvalues are complex (which is the case
already for fairly small non-Hermiticity parameters) the nearest-neighbour
distances of the (unfolded) quasienergies follow a two-dimensional Posisson
distribution when the classical dynamics is regular. In the chaotic regime,
they are in line with recently identified universal complex level spacing
distributions for non-Hermitian systems, with transpose symmetry $A^T= A$. It
is demonstrated how breaking this symmetry (by introducing an extra term in the
Hamiltonian) recovers the more familiar universality class for non-Hermitian
systems given by the complex Ginibre ensemble. Both universality classes
display cubic level repulsion. The PT-symmetry of the system does not seem to
influence the complex level spacings. Similar behaviour is also observed for
the spectrum of a PT-symmetric extension of the triadic Baker map.
