mixmaster regime is unfortunately – and perhaps not surprisingly – not so well developed. Rigorous results are scarce (note [@mixinhom]) and even in the case of the spatially homogeneous Bianchi IX model only partial results exist in the literature [@mix].

The purpose of this paper is to extend the Fuchsian approach of [@AR] to the more complicated class of models described by the action (\[001\]) and to prove that those among the above models that were predicted in [@DHS; @dh1; @dh2] to be Kasner-like are indeed so. This provides many new instances where one can rigorously construct a general singular solution for a coupled Einstein-matter (or pure Einstein, in $D \geq 11$) system. In fact, our (Fuchsian-system-based) results prove that the formal perturbative solutions that can be explicitly built for these models do converge to exact solutions. This provides a further confirmation of the general validity of the BKL ideas. We shall also explicitly determine the subcritical domain $\cal D$ for a few illustrative models. For all the relevant systems, we construct local (near the singularity) analytic solutions, which are “general” in the sense that they contain the right number of freely adjustable arbitrary functions of space (in particular, these 