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 $G(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})\mathrm{/}G[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$, given by quotients of the double loop group $G(\text{\hspace{0.17em}⁣}(x)\text{\hspace{0.17em}⁣})(\text{\hspace{0.17em}⁣}(y)\text{\hspace{0.17em}⁣})$. 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.