  We present an algebraic method to study four-dimensional toric varieties by
lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb
Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. 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 and which are in the same
twisting index class. In particular, we show that any semitoric system with
precisely one focus-focus singular point can be continuously deformed into a
system in the same isomorphism class as the Jaynes-Cummings model from optics.
