  In this paper, we use algebro-geometric methods in order to derive
classification results for so-called $D$-bialgebra structures on the power
series algebra $A[\![z]\!]$ for certain central simple non-associative algebras
$A$. These structures are closely related to a version of the classical
Yang-Baxter equation (CYBE) over $A$.
  If $A$ is a Lie algebra, we obtain new proofs for pivotal steps in the known
classification of non-degenerate topological Lie bialgebra structures on
$A[\![z]\!]$ as well as of non-degenerate solutions of the usual CYBE.
  If $A$ is associative, we achieve the classification of non-triangular
topological balanced infinitesimal bialgebra structures on $A[\![z]\!]$ as well
as of all non-degenerate solutions of an associative version of the CYBE.
