On zero-curvature representations of partial differential equations M. Marvan Abstract. This paper concerns the problem of finding all possible zero-curvature representations of a given system of partial differential equations. A new computational procedure is proposed, designed to give complete answers. The procedure is applicable to non-overdetermined systems when the number $m$ of independent variables is two, while for $m > 3$ we prove a theorem of non-existence. The approach, inspired by A.M.~Vinogradov's $\C$-spectral sequence, is based on finding representatives of gauge equivalence classes. Keywords. Nonlinear partial differential equation, zero-curvature representation, linear pseudopotential, linear covering, AKNS formulation, diffiety, $\C$-spectral sequence, gauge complex, gauge equivalence, jet comonad. MS classification numbers. 35A30, 58G05; 35G20, 18C15.