  In this work, we apply the tools of the dynamical system theory in order to
revisit and uncover the structure of a nongravitational interaction between
pressureless dark matter and dark energy described by a scalar field, which has
been previously investigated in the literature. For a coupling function $Q =
-(\alpha \dot{\rho}_m + \beta \dot{\rho}_{\phi} )$, we have found that it can
be rewritten in the form $Q = 3H (\alpha \rho_m + \beta \dot{\phi}^2)/(1-\alpha
+\beta)$, so that its dependence on the dark matter density and on the kinetic
term of the scalar field is linear and proportional to the Hubble parameter. We
analyze the following scenarios $\alpha=0$, $\alpha = \beta$ and $\alpha =
-\beta$, separately and in order to describe the cosmological evolution for
each solution we have calculated various observables. We find that there are
not any new stable late-time solutions apart from those found of standard
quintessence, nevertheless, the stability conditions are severely altered. A
notable result found with respect to previous works is that in our case, with
the exception of the matter dominated solution, the remaining critical points
behave as scaling although the stiff matter solution and the dark energy
dominated state can be recovered in the limit $\beta \rightarrow 0$ and $\beta
\rightarrow 1$, respectively. Moreover, it is shown that for $\alpha = \beta $
and $\alpha = - \beta$ (in general for $\alpha \neq 0$), a separatrix arises
modifying prominently the structure of the phase space. This represents a novel
feature no mentioned before in the literature.
