An algebraic classification of solution generating techniques

We consider a two-fold problem: on the one hand, classification of family solution-generating techniques in (modified) supergravity and, other canonical transformations 2-dimensional $\sigma$-models giving rise to integrable-preserving transformations. Assuming generalised Scherk-Schwarz ansatz, fact, two problems admit essentially same algebraic formulation, emerging from an underlying double Lie algebra $\mathfrak d$. After presenting our derivation classification, we discuss detail relation modified and additional conditions recover standard (unmodified) supergravity. Starting master equation - that encodes all possible continuous deformations allowed show these are classified by cohomologies $H^2(\mathfrak h,\mathbb R)$ $H^3(\mathfrak maximally isotropic subalgebra h$ {We illustrate results with non-trivial example, bi-Yang-Baxter-Wess-Zumino model.