above statements were derived by adopting a line of thought analogous to that followed by BKL. Now, as understood by BKL themselves, these arguments, although quite convincing, are somewhat heuristic. It is true that the original arguments have received since then a considerable amount of both numerical and analytical support [@bgimw; @BIW; @U1; @WIB; @garf]. Yet, they still await a complete proof. One notable exception is the four-dimensional gravity-dilaton system, which has been rigorously demonstrated in [@AR] to be indeed Kasner-like, confirming the original analysis [@BK2]. Using Fuchsian techniques, the authors of [@AR] have proven the existence of a local (analytic) Kasner-like solution to the Einstein-dilaton equations in four dimensions that contains as many arbitrary, physically relevant functions of space as there are local degrees of freedom, namely $6$ (counting $q$ and $\dot{q}$ independently). To our knowledge, this was the first construction, in a rigorous mathematical sense, of a general singular solution for a coupled Einstein-matter system. Note in this respect several previous works in which formal solutions had been constructed near (Kasner-like) cosmological singularities by explicit perturbative methods, to all orders of perturbation theory [@GM93; @BDV].

The situation concerning the more complicated (and in some sense more interesting) generalized 