**Authors:** Bujac Cristina

### Abstract

In this article we classify a subfamily of differential
real cubic systems possessing eight invariant straight lines,
including the line at infinity and including their multiplicities.
This subfamily of systems is characterized by the existence of
two distinct infinite singularities, defined by the linear
factors of the polynomial\ $ C_3(x,y)=yp_3(x,y)-xq_3(x,y)$,\ where
$p_3$ and $q_3$ are the cubic homogeneities of these systems.
Moreover we impose additional conditions related with the
existence of triplets and/or couples of parallel invariant lines.
This classification, which is taken modulo the action of the group
of real affine transformations and time rescaling, is given in
terms of affine invariant polynomials. The invariant polynomials
allow one to verify for any given real cubic system whether or not
it has invariant straight lines of total multiplicity eight, and
to specify its configuration of straight lines endowed with their
corresponding real singularities of this system. The calculations
can be implemented on computer and the results can therefore be
applied for any family of cubic systems in this class, given in
any normal form.

Institute of Mathematics and Computer Science

Academy of Sciences of Moldova

E-mail:

### Fulltext

–

0.58 Mb