Topological Manin pairs and $$(n,s)$$-type series

Lie subalgebras of $ L = \mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^n\mathfrak{g}[x] $, complementary to the diagonal embedding $\Delta$ \mathfrak{g}[\![x]\!] and Lagrangian with respect some particular form, are in bijection formal classical $r$-matrices topological bialgebra structures on algebra power series $. In this work we consider arbitrary subspaces associate them so-called type (n,s) We prove that skew-symmetric $-type quasi-Lie Using classificaiton Manin pairs classify up twisting coordinate transformations all structures. Series solving generalized Yang-Baxter equation, correspond $L$. discuss their possible utility theory integrable systems.