Methods of Nonassociative Algebras in Differential Equations

Many well-known (classes of) differential equations  may be viewed as an equation in a certain
commutative nonassociative algebra. We develop further the principal idea of L. Markus for deriving
algebraic properties of solutions to ODEs and PDEs directly from the equations defining them.
Our main purpose is a) to show how the algebraic formalism can be applied with great success to a
remarkably elegant description of the geometry of curves being solutions to homogeneous polynomial ODEs,
and, on the other hand, b) to motivate the recent interest in applications of nonassociative algebra
methods to PDEs. More precisely, given a differential equation on an algebra A, we are interested in
the following two problems:

1. Which properties of the differential equation determine certain algebraic structures on $A$ such as
to be power associative, unital or division algebra.

2. In the converse direction, which properties of $A$ imply certain qualitative information about the
differential equation, for example topological equivalent classes, existence of a bounded, periodic,
ray solutions, ellipticity, etc.

We also define and discuss syzygies between Peirce numbers which provide an effective tool for our study.
(Some results here are based on a recent joint work with V. Tkachev.)