The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies

We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular $\mathbb Z{\times}\mathbb Z$ matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker–Akhiezer function.