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 Grassmannians $Gr_{\mathbb
R}(k,N)$ and $Gr_{\mathbb C}(k,N)$. We proved the regularity of all
solutions obtained