  A practical search technique for finding the complex saddle points used in
wave packet or coherent state propagation is developed which works for a large
class of Hamiltonian dynamical systems with many degrees of freedom. The method
can be applied to problems in atomic, molecular, and optical physics, and other
domains. A Bose-Hubbard model is used to illustrate the application to a
many-body system where discrete symmetries play an important and fascinating
role. For multidimensional wave packet propagation, locating the necessary
saddles involves the seemingly insurmountable difficulty of solving a boundary
value problem in a high-dimensional complex space, followed by determining
whether each particular saddle found actually contributes. In principle, this
must be done for each propagation time considered. The method derived here
identifies a real search space of minimal dimension, which leads to a complete
set of contributing saddles up to intermediate times much longer than the
Ehrenfest time scale for the system. The analysis also gives a powerful tool
for rapidly identifying the various dynamical regimes of the system.
