  We consider a system of three commuting difference operators in three
variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$.
This system and its eigenfunctions generalize the trigonometric $A_1$
Ruijsenaars-Schneider model and $A_1$ Macdonald polynomials, respectively. The
principal object of study in this paper is the algebra generated by these
difference operators together with operators of multiplication by $x_{ij} +
x_{ij}^{-1}$. We represent the Dehn twists by outer automorphisms of this
algebra and prove that these automorphisms satisfy all relations of the mapping
class group of the closed genus two surface. Therefore we argue from
topological perspective this algebra is a genus two generalization of $A_1$
spherical DAHA.
