  In this paper we study families of Lagrangian tori that appear in a
neighborhood of a resonance of a near-integrable Hamiltonian system. Such
families disappear in the "integrable" limit $\varepsilon\to 0$. Dynamics on
these tori is oscillatory in the direction of the resonance phases and rotating
with respect to the other (non-resonant) phases.
  We also show that, if multiplicity of the resonance equals one, generically
these tori occupy a set of large relative measure in the resonant domains in
the sense that the relative measure of the remaining "chaotic" set is of order
$\sqrt\varepsilon$. Therefore for small $\varepsilon > 0$ a random initial
condition in a $\sqrt\varepsilon$-neighborhood of a single resonance occurs
inside this set (and therefore generates a quasi-periodic motion) with a
probability much larger than in the "chaotic" set.
  We present results of numerical simulations and discuss the form of
projection of such tori to the action space.

Dynamical Systems