of MW and M31.

This work extends a previous paper (Del Popolo [*et al.*]{} 2014), enriched with the part of the model described in appendix \[Dynamics of the satellites\], and will chiefly focus on the latter problem (MSP). However, the model also carries the solution for the former two (CCP and AMC), from the part of the model developed in Del Popolo [*et al.*]{} (2014). In clear, it uses a semi-analytical model to account for the dynamical evolution of satellites. The model, originated in DP09 (and DP12a, b), is an improved spherical infall model already discussed by many authors (Gunn & Gott 1972; Fillmore & Goldreich 1984; Bertschinger 1985; Hoffman & Shaham 1985; Ryden & Gunn 1987; Henriksen & Widrow 1995, 1997, 1999; Henriksen & Le Delliou 2002; Le Delliou & Henriksen 2003; Le Delliou 2008; Ascasibar, Yepes & Gottlöber 2004; Williams, Babul & Dalcanton 2004; Le Delliou, Henriksen & MacMillan 2010, 2011a, 2011b)[^6].

In the present paper, we follow the path opened by Z12 and B13, but we consider another mechanism than SF that is also better able to flatten the density profiles of satellites. Namely, we use a mechanism based on the exchange of energy and angular momentum 