In classical electrodynamics on a simply connected space, equation $dF=0$ is equivalent to the existence of an electrodynamic potential (standard Poincaré lemma). From a mathematical point of view, de Rham complex (algebra of total antisymmetric tensor fields with operator $d$ fulfilling $d^2=0$) is the foundation of classical electrodynamics. In other tensor field theories we work with a field T of more complicated symmetry and one of the field equations is $DT=0$, where $D$ is some first order differential operator. In this talk I will define an $N$-complex (algebra of tensor fields of given Young symmetry with operator $D$ fulfilling $D^N=0$) and a generalized Poincaré lemma, which is the reason for the existence of a field potential in every integer-spin field theory.