  In many problems of quantum chaos the calculation of sums of products of
periodic orbit contributions is required. A general method of computation of
these sums is proposed for generic integrable models where the summation over
periodic orbits is reduced to the summation over integer vectors uniquely
associated with periodic orbits. It is demonstrated that in multiple sums over
such integer vectors there exist hidden saddle points which permit explicit
evaluation of these sums. Saddle point manifolds consist of periodic orbits
vectors which are almost mutually parallel. Different problems has been treated
by this saddle point method, e.g. Berry's bootstrap relations, mean values of
Green function products etc. In particular, it is obtained that suitably
defined 2-point correlation form-factor for periodic orbit actions in generic
integrable models is proportional to quantum density of states and has peaks at
quantum eigenenergies.
