Strong integrability of $\lambda$-deformed models

We study the notion of strong integrability for classically integrable $\lambda$-deformed CFTs and coset CFTs. To achieve this goal we employ the Poisson brackets of the spatial Lax matrix which we prove that it assumes the Maillet $r\mathrm{/}s$-matrix algebra. As a consequence the system in question are integrable in the strong sense. Furthermore, we show that the derived Maillet $r\mathrm{/}s$-matrix algebras can be realized in terms of twist functions, at the poles of which we recover the underlying symmetry algebras.