Generalized classical Yang-Baxter equation and regular decompositions

The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra decompositions of $\mathfrak{g}(\text{\hspace{0.17em}⁣}(x)\text{\hspace{0.17em}⁣})\times \mathfrak{g}[x]\mathrm{/}{x}^m\mathfrak{g}[x]$. The latter decompositions are in bijection with the solutions to the GCYBE. Under appropriate regularity conditions, we obtain a partial classification of such solutions. The paper is concluded with the presentations of the Gaudin-type models associated to these solutions.