*(English)*Zbl 0791.17002

Consider a differential equation ${x}^{\text{'}}=p\left(x\right)$, with $p$ homogeneous quadratic, on a finite dimensional vector space $V$. Define a bilinear composition on $V$ by $xy=\frac{1}{2}(p(x+y)-p\left(x\right)-p\left(y\right))$, Then $V$ with this multiplication, is a commutative, nonassociative algebra, and a linear, solution-preserving map between two differential equations of the type above is a homomorphism of the corresponding algebras.

The first part (1-6) contains the fundamental theory and its application to some classes of examples. The basic definitions and properties are presented; then subalgebras and algebraic invariant subsets are studied with emphasis on semi-invariants.

In the second part (7-10), the objects of investigation are subalgebras of the Lie algebra Pol $V$ of all polynomial vector fields on a vector space $V$. If a given homogeneous polynomial $p$ is contained in a large finite dimensional subalgebra of Pol $V$ or if its centralizer is large, it implies conditions on the algebra determined by $p$ that can be very strong. This connection between the inner structure of $p$ and its role in Pol $V$ is the main subject of interest. The transitive subalgebra and the differential equations related to it are discussed, and the centralizer of a given homogeneous element of Pol $V$ is studied. Classes of differential equations can thus be solved explicitly.

##### MSC:

17A01 | General theory of nonassociative algebra |

17-01 | Textbooks (nonassociative rings and algebras) |

34-01 | Textbooks (ordinary differential equations) |

34A34 | Nonlinear ODE and systems, general |

34C20 | Transformation and reduction of ODE and systems, normal forms |

17B66 | Lie algebras of vector fields and related (super)algebras |

17A36 | Automorphisms, derivations, other operators |