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Twisted Teichmüller curves

Let $X_D$ denote the Hilbert modular surface $\HH \times \HH^- / \SL_2(\OD)$. In \cite{HZ76}, F. Hirzebruch and D. Zagier introduced Hirzebruch-Zagier cycles, that could also be called twisted diagonals. These are maps $\HH \to \HH \times \HH^-$ given by $z \mapsto (Mz,-M^σz)$ where $M \in \GL_2^+(K)$ and $σ$ denotes the Galois conjugate. The projection of a twisted diagonal to $X_D$ yields a Kobayashi curve, i.e. an algebraic curve which is a geodesic for the Kobayashi metric on $X_D$. Properties of Hirzebruch-Zagier cycles have been abundantly studied in the literature.\\Teichmüller curves are algebraic curves in the moduli space of Riemann surfaces $\mathcal{M}_g$, which are geodesic for the Kobayashi metric. Some Teichmüller curves in $\mathcal{M}_2$, namely the primitive ones, can also be regarded as Kobayashi curves on $X_D$. This implies that in the universal cover the curve is of the form $z \mapsto (z,φ(z))$ for some holomorphic map $φ$. A possibility to construct even more Kobayashi curves on $X_D$ is to consider the projection of $(Mz,M^σφ(z))$ to $X_D$ where again $M \in \GL_2^+(K)$. These new objects are called twisted Teichmüller curves because their construction remi