We develop a new asymptotic method of resolution of the two-dimensional equilibrium equation of collisionless plasmas described by the Maxwell–Vlasov equations. This method differs from the classical one proposed by K. Schindler [Earth’s Magnetospheric Processes (ed. B. M. McCormac). Norwood, MA: Reidel, 1972, pp. 200–209.] since we consider free-boundary plasmas. Our method is a generalization of the usual multiscale asymptotic developments. The first-approximation asymptotic solutions are found from the elimination of increasing and singular terms in the next approximation. We apply the method to the mathematical description of nonlinear structures that may form in neutral sheets. Particular solutions describing localized plasmoids (O-point configuration) as well as X-point magnetic configurations are obtained. We also find more general solutions describing a finite number of ‘magnetic islands’ (multiplasmoid solutions) separated by X points.