  Under conditions of clean flow we compute the leading term in the STF when
the set of periods of the energy surface is discrete. Comparing to the case of
non-degenerate periodic orbits, we obtain a supplementary term which is given
in terms of the linearized flow. As particular cases, we give a STF for
quadratic Hamiltonians and we obtain the Berry-Tabor formula for integrable
systems. For conservative systems (i.e. systems with several first integrals),
we give practical conditions to get a clean flow and interpret the leading term
of the STF for a compact symmetry. We give several examples to illustrate our
computation.
