The sequence of equilibria of two-dimensional reduced magnetohydrodynamics has been studied as a function of the tearing mode stability parameter . After a symmetry-breaking bifurcation occurring at , which originates a state with a small magnetic island, the system undergoes a second bifurcation, of tangent type, at . Above this value, no stationary solutions with small islands exist. The system rapidly develops an island of macroscopic size. This general property is proposed as a basic ingredient of the intermittent events observed in magnetically confined plasmas.