  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 $\mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^m
\mathfrak{g}[x]$. 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.
