  For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated
to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs
bundles on a Riemann surface C agrees with the weight filtration on the
cohomology of the twisted G character variety of C, when the cohomologies are
identified via non-Abelian Hodge theory. The proof is accomplished by means of
a study of the topology of the Hitchin map over the locus of integral spectral
curves.
