Totally Real Minimal Tori in $CP^2$

In this paper we show that all totally real superconformal minimal tori in $CP^2$ correspond with doubly-periodic finite gap solutions of the Tzitzeica equation $$\omega_{z{\overline z}}=e^{-2\omega}-e^{\omega}$$ Using the results on the Tzitzeica equation in integrable system theory, we describe explicitly all these tori by Prym-theta functions.