  We suggest a new scheme of derivation of a non-linear ballistic
$\sigma$-model for a long range disorder and quantum billiards. The derivation
is based on writing equations for quasiclassical Green functions for a fixed
long range potential and exact representation of their solutions in terms of
functional integrals over supermatrices $Q$ with the constraint $Q^2=1$.
Averaging over the long range disorder or energy we are able to write a
ballistic $\sigma$-model for all distances exceeding the electron wavelength.
Neither singling out slow modes nor a saddle-point approximation are used in
the derivation. Carrying out a course graining procedure that allows us to get
rid off scales in the Lapunov region we come to a reduced $\sigma$-model
containing a conventional collision term. For quantum billiards, we demonstrate
that, at not very low frequencies, one can reduce the $\sigma$-model to a
one-dimensional $\sigma$-model on periodic orbits. Solving the latter model,
first approximately and then exactly, we resolve the problem of repetitions.
