Classifying toric and semitoric fans by lifting equations from $\textrm{SL}_2 (\mathbb{Z})$

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $\textrm{SL}_2 (\mathbb{Z})$ to its preimage in the universal cover of $\textrm{SL}_2 (\mathbb{R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into the Jaynes-Cummings model from optics.