  In their generalization of the rational homotopy theory to non-simply
connected spaces, G\'omez-Tato--Halperin--Tanr\'e adopted local systems of
commutative differential graded algebras (CDGA's) as algebraic models. As
another non-simply connected rational model, a $\pi_1$-equivariant CDGA with a
semi-simple action was introduced by Hain and studied by
Katzarkov-Pantev-To\"en and Pridham. These models have their own advantages and
direct comparison might be important. In this note, we present an example of
local system of CDGA's which is a non-nilpotent version of an example given by
G\'omez-Tato et al., and compute the corresponding $\pi_1$-equivariant CDGA. We
also see how some homotopy invariants such as homotopy groups and twisted
cohomology are algebraically recovered from the equivariant CDGA with this
example. We also describe a general procedure to obtain the corresponding
equivariant CDGA model from a local system model in the case of
$\pi_1=\mathbb{Z}\times \mathbb{Z}$.
