  The analysis of dynamical systems has been a topic of great interest for
researches mathematical sciences for a long times. The implementation of
several devices and tools have been useful in the finding of solutions as well
to describe common behaviors of parametric families of these systems. In this
paper we study deeply a particular parametric family of differential equations,
the so-called \emph{Linear Polyanin-Zaitsev Vector Field}, which has been
introduced in a general case, in the previous paper by the same authors, as a
correction of a family presented in the classical book of Polyanin and Zaitsev.
Linear Polyanin-Zaitsev Vector Field is transformed into a Li\'enard equation
and in particular we obtain the Van Der Pol equation. We present some algebraic
and qualitative results to illustrate some interactions between algebra and the
qualitative theory of differential equations in this parametric family.
