In this work, we analyse the sequence of bifurcated equilibria in two-dimensional reduced magnetohydrodynamics. Diamagnetic effects are studied under the assumption of a constant equilibrium pressure gradient, not altered by the formation of a magnetic island. The formation of an island when the symmetric equilibrium becomes unstable is studied as a function of the tearing-mode stability parameter Δ′ and of the diamagnetic frequency, by employing fixed-point numerical technique and an initial-value code. At larger values of Δ′, a tangent bifurcation takes place, above which no small-island solutions exist. This bifurcation persists up to fairly large values of the diamagnetic frequency (of the order of one-tenth of the Alfvén frequency). The implications of this phenomenology for the intermittent MHD dynamics observed in tokamaks is discussed.