Two dimensional versions of the affine Grassmannian and their geometric description

For a smooth affine algebraic group G over an algebraically closed field, we consider several two-variables generalizations of the affine Grassmannian G((t))/G[[t]], given by quotients of the double loop group G((x))((y)). We prove that they are representable by ind-schemes if G is solvable. Given a smooth surface X and a flag of subschemes of X, we provide a geometric interpretation of the two-variables Grassmannians, in terms of bundles and trivialisation data defined on appropriate loci in X, which depend on the flag.