Newton method may serve as a tool for solution of underdetermined systems of algebraic (differentiable) equations P(x) = 0, P : Rn -> Rm , m < n. It is usually written via pseudo-inverse matrix, which correspond to Euclidean norms in pre-image and image spaces [1, 4]. The same method can be used to explore image set of a non-linear differentiable mapping g(x) , resulting in equations of type g(x) = c y, with chosen direction y. We propose variable-norm setup for Newton method. Using generic convergence conditions, based on technique of [2, 3] we study different norm combinations, choice of norms for image exploration problems, as well as constant estimation issues related with norm choice.