Zero-curvature representations (ZCRs) \[ D_t A- D_x B-[A,B ]=0 \] of evolution equations \(u_t= f(x, u, u_1, \dots, u_n)\) are investigated in this paper. It is shown that the existence condition of the ZCR with matrices \(A= A(x, u, u_1, \dots, u_i)\) and \(B= B(x, u, u_1, \dots,u_{i+n -1})\) can be represented in the form \[ fC= \nabla P, \] where \(C= \sum_{k=1}^i (-\nabla)^k (\partial_k A)\), \(P= B- \sum_{k=0}^i \sum_{l=1}^k (D_x^{k-l} f) (-\nabla )^{l-1} (l_k A)\) are tensors under gauge transformations \(A'= SAS^{-1}- (D_x S)S^{-1}\), \(B'= SBS^{-1}- (D_t S)S^{-1}\) and \(\nabla= D_x+ [A, \cdot ]\). It is proved that every matrix \(A(x, u, u_1, \dots)\) determines a continual class of equations \[ u_t= (D_x^r+ a_{r-1} D_x^{r-1}+ \dots+ a_1 D_x+ a_0) p+ \varphi_1 q_1+ \dots+ \varphi_s q_s, \] admitting ZCRs with this \(A\); here the function \(p(x, u, u_1, \dots)\) and the constants \(\varphi_1, \dots, \varphi_s\) are arbitrary whereas the integers \(r\) and \(s\) and the functions \(a_0, \dots, a_{r-1}\), \(q_1, \dots, q_s\) of \(x, u, u_1, \dots\) are explicitly determined by \(A\). A dependence of this class on an essential parameter of \(A\) is illustrated by examples.