The singular fibre of the Hitchin map

Given any line bundle L of positive degree, on a compact Riemann surface, let $M_L^\Lambda$ be the moduli space of L-twisted Higgs pairs of rank 2 with fixed determinant isomorphic to $\Lambda$ and traceless Higgs field. We give a description of the singular fibre of the Hitchin map $H:M^L_\Lambda\to H^0(L^2)$, when the corresponding spectral curve has any singularity of type $A_{m-1}$. In particular, we prove directly that this fibre is connected.