  We consider the problem of multiparticle production at threshold in a
$\phi^4$-theory with an $O(N_1$$+$$N_2)$ symmetry softly broken down to
$O(N_1)\times O(N_2)$ by nonequal masses. We derive the set of recurrence
relations between the multiparticle amplitudes which sums all relevant diagrams
with arbitrary number of loops in the large-$N$ limit with fixed number of
produced particles. We transform it into a quantum mechanical problem and show
how it can be obtained directly from the operator equations of motion by
applying the factorization at large $N$. We find the exact solutions to the
problem by using the Gelfand--Diki\u{\i} representation of the diagonal
resolvent of the Schr\"{o}dinger operator. The result coincides with the tree
amplitudes while the effect of loops is the renormalization of the coupling
constant and masses. The form of the solution is due to the fact that the exact
amplitude of the process $2$$\ra$$n$ vanishes at $n$$>$$2$ on mass shell when
averaged over the $O(N_{1,2})$-indices of incoming particles. We discuss what
dynamical symmetry is behind this property. We also give an exact solution in
the large-$N$ limit for the model of the $O(N)$$+$$singlet$ scalar particle
with the spontaneous breaking of a reflection symmetry.
