Integrable $(2+1)$-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable $(2+1)$-dimensional systems of hydrodynamic type. The Gibbons–Tsarev (GT) systems are most fundamental here. A whole class of integrable $(2+1)$-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus $g=0$ and $g=1$ and also a new GT system corresponding to algebraic curves of genus $g=2$. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial”.