$\cal D$, [*i.e.,*]{} the critical value $\lambda_c$ such that the system is Kasner-like when $-\lambda_c < \lambda < \lambda_c$. Because this case is exemplary of the general situation, while still being technically rather simple to handle, we shall describe in some detail the explicit steps of the Fuchsian approach.

In section \[vacuum\], vacuum solutions governed by the pure Einstein action (\[action2\]) with $D \geq 11$ are considered. This system was argued in [@DHS] to be Kasner-like and we show here how this rigorously follows from the Fuchsian approach. Note that, contrary to what happens when a dilaton is present, Fuchsian techniques apply here even though not all Kasner exponents can be positive.

In sections \[scalar\] – \[yangmills\], the results of the previous sections are generalized to the wider class of systems (\[001\]). First, in section \[scalar\], to solutions of Einstein’s equation with spacetime dimension $D \geq 3$ and a matter source consisting of a massless scalar field, governed by the action $S_E [g_{\alpha \beta}] + S_\phi[g_{\alpha \beta}, \phi]$. This is the generalization to any $D \geq 3$ of the case $D = 4$ treated in [@AR]. In section \[nform\], we turn to the general situation described by the action (\[001\]), 