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We study the physical and evolutionary properties of the `white dwarf (WD) family' of AM CVn stars by computing realistic models of interacting double-degenerate systems. We evaluate self-consistently both the mass-transfer rate from the donor, as determined by gravitational wave emission and interaction with the binary companion, and the thermal response of the accretor to mass deposition. We find that, after the onset of mass transfer, all the considered systems undergo a strong non-dynamical He-flash. However, due to the compactness of these systems, the expanding accretors fill their Roche lobe very soon, thus preventing the efficient heating of the external layers of the accreted CO WDs. Moreover, due to the loss of matter from the systems, the orbital separations enlarge and mass transfer comes to a halt. The further evolution depends on the value of 3{dot}M after the donors fill again their lobe. On one hand, if the accretion rate, as determined by the actual value of (Mdon, Macc), is high enough, the accretors experience several He-flashes of decreasing strength and then quiescent He-burning sets in. Later on, since the mass-transfer rate in IDD is a permanently decreasing function of time, accretors experience several recurrent strong flashes. On the other hand, for intermediate and low values of 3{dot}M the accretors enter directly the strong flashes accretion regime. As expected, in all the considered systems the last He-flash is the strongest one, even if the physical conditions suitable for a dynamical event are never attained. When the mass accretion rate decreases below (2-3)x10^–8 M_☉/yr, the compressional heating of the He-shell becomes less efficient than the neutrino cooling, so that all the accretors in the considered systems evolve into massive degenerate objects. Our results suggest that SNe.Ia or Type Ia Supernovae due to Edge-Lit Detonation in the WD family of AM CVn stars should be much more rare than previously expected.