**Abstract.**
The quadratizations of a (homogeneous nonquadratic) nonlinear polynomial
system of ODEs introduced by Myung and Sagle in [17] is
considered. The 1-1 correspondence between homogeneous quadratic systems of
ODEs and nonassociative algebras is used to prove a special
structure of the algebra corresponding to a general homogeneous quadratic
systems being a quadratization. Every homogeneous solution-preserving map
(corresponding to a quadratization) determines the so called essential set
which turns out to be crucial for preserving the (in)stability of the origin
from homogeneous nonquadratic systems to their quadratizations and vice versa.
In particular the quadratizations of homogeneous systems *x' =f _{α}(x) *
(of order

**2010 Mathematics Subject Classification.**
34A34, 34D20, 13P99.

**Key words and phrases.** Homogeneous system, cubic system, quadratic system,
quadratization, commutative (nonassociative) algebra, stability, critical point.

DOI: 10.3336/gm.50.1.09

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