  We study Veech surfaces of genus 2 arising from quadratic differentials that
are not squares of abelian differentials. We prove that all such surfaces of
type (2,2) and (2,1,1) are arithmetic. In (1,1,1,1) case, we reduce the
question to abelian differentials of type (2,2) on hyperelliptic genus 3
surfaces with singularities at Weierstrass points, and we give an example of a
non-arithmetic Veech surface.
