Algebraic integrability of the two-body Ruijsenaars operator
  We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem. 
Giovanni Felder (ETH Zuerich), Alexander Varchenko (UNC Chapel Hill)