  It is explored a model of compact Riemann surfaces in genus two, represented
geometrically by two-parametric hyperbolic octagons with an order four
automorphism. We compute the generators of associated isometry group and give a
real-analytic description of corresponding Teichm\"uller space, parametrized by
the Fenchel-Nielsen variables, in terms of geometric data. We state the
structure of parameter space by computing the Weil-Petersson symplectic
two-form and analyzing the isoperimetric orbits. The results of the paper may
be interesting due to their applications to the quantum geometry, chaotic
systems and low-dimensional gravity.
