equations of the form $t \partial_t u^{(n)} + {\cal A}(x)  u^{(n)} =
f(t,x,u^{(n-1)},u^{(n-1)}_x)$. At each step in this iterative process the source term is a known function which essentially behaves (modulo logarithms) like a sum of powers of $t$ (with space-dependent coefficients). The crucial step in the iteration is then to solve equations of the type $t \partial_t u + {\cal A}(x)  u = C(x) t^{\delta(x)}$. The positivity condition on ${\cal A}(x)$ guarantees the absence of homogeneous solutions remaining bounded as $ t \rightarrow 0$, and ensures the absence of “small denominators” in the (unique bounded) inhomogeneous solution generated by each partial source term: $ u_{\rm inhom} = (\delta + {\cal A})^{-1} C t^{\delta}$. (See, [*e.g.,*]{} [@BDV] for a concrete iterative construction of a Kasner-like solution and the proof that it extends to all orders.) This link between Fuchsian systems and “good systems” that can be solved to all orders in a formal iteration makes it a priori probable that all cases which the heuristic approach à la BKL has shown to be asymptotic to a Kasner-like solution (by checking that the leading “post-Kasner” contribution is asymptotically sub-dominant) can be cast in a Fuchsian form. The main technical burden of 