A cyclic cover of the projective plane branched at four points has a natural
structure of a square-tiled surface. We describe the combinatorics of such a
square-tiled surface, the geometry of the corresponding Teichm\"uller curve,
and compute the Lyapunov exponents of the determinant bundle over the
Teichm\"uller curve with respect to the geodesic flow.

We find a new example of a Teichm\"uller curve with a completely degenerate
Lyapunov spectrum (the only known example found previously by G. Forni also
corresponds to a cyclic cover). Presumably, these two examples cover all
possible Teichm\"uller curves with completely degenerate Lyapunov spectrum.