In this manuscript, we design an efficient sixth-order scheme for solving nonlinear systems of equations, with only two steps in its iterative expression. Moreover, it belongs to a new parametric class of methods whose order of convergence is, at least, four. In this family, the most stable members have been selected by using techniques of real multidimensional dynamics; also, some members with undesirable chaotic behavior have been found and rejected for practical purposes. Finally, all these high-order schemes have been numerically checked and compared with other existing procedures of the same order of convergence, showing good and stable performance.