  We study the one parameter family of genus 2 Riemann surfaces defined by the
orbit of the L-shaped translation surface tiled by three squares under the
Teichm\"uller geodesic flow. These surfaces are real algebraic curves with
three real components. We are interested in describing these surfaces by their
period matrices. We show that the only Riemann surface in that family admitting
a non-hyperelliptic automorphism comes from the 3-square-tiled translation
surface itself. This makes the computation of an exact expression for period
matrices of other Riemann surfaces in that family by the classical method
impossible. We nevertheless give the solution to the Schottky problem for that
family: we exhibit explicit necessary and sufficient conditions for a Riemann
matrix to be a period matrix of a Riemann surface in the family, involving the
vanishing of a genus 3 theta characteristic on a family of double covers.
