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

Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system