  In the paper we develop the dressing method for the solution of the
two-dimensional periodic Volterra system with a period N. We derive soliton
solutions of arbitrary rank $k$ and give a full classification of rank 1
solutions. We have found a new class of exact solutions corresponding to wave
fronts which represent smooth interfaces between two nonlinear periodic waves
or a periodic wave and a trivial (zero) solution. The wave fronts are
non-stationary and they propagate with a constant average velocity. The system
also has soliton solutions similar to breathers, which resembles soliton webs
in the KP theory. We associate the classification of soliton solutions with the
Schubert decomposition of the Grassmanians ${\rm Gr}_{\mathbb{R}}(k,N)$ and
${\rm Gr}_{\mathbb{C}}(k,N)$.
