A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval [0, T) solution to the Cauchy-Neumann problem is studied. For the situation when the "local energies" of the solution are uniformly bounded on [0, T), smooth extendibility of the solution up to t = T is proved. In the case when [0, T) defines the maximal interval of the existence of a smooth solution, the singular set at the moment t = T is described.