  In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described
the asymptotic behaviour of trajectories of the full symmetric
$\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax
matrix. It turned out that it is completely determined by the Bruhat order on
the permutation group. In the present paper we extend this result to the case
when some eigenvalues of the Lax matrix coincide. In that case the trajectories
are described in terms of the projection to a partial flag space where the
induced dynamical system verifies the same properties as before: we show that
when $t\to\pm\infty$ the trajectories of the induced dynamical system converge
to a finite set of points in the partial flag space indexed by the Schubert
cells so that any two points of this set are connected by a trajectory if and
only if the corresponding cells are adjacent. This relation can be explained in
terms of the Bruhat order on multiset permutations.
