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A generalized auxiliary equation method is proposed to construct more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including soliton-like solutions, triangular-like solitions, single and combined non-degenerate Jacobi elliptic doubly-like periodic solutions, and Weierstrass elliptic doubly-like periodic solutions.