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 $alpha eq 0$), a separatrix arises modifying prominently the structure of the phase space. This represents a novel feature no mentioned before in the literature.
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