the present work will indeed be to show in detail how this can be carried out for the evolution systems corresponding to all the sub-critical ([*i.e.,*]{} non-oscillatory) Einstein-matter systems. Our Fuchsian formulation proves that (in the analytic case) the formal all-orders iterative solutions for the models we consider do actually converge to the unique, exact solution having a given leading Kasner asymptotic behaviour as $ t \to 0$.

Finally, the fourth step of our strategy is to prove that the constructed solution does satisfy also all the Einstein and Gauss-like constraints so that it is a solution of the full set of Einstein-matter equations. We shall deal successively with the matter (Gauss-like) constraints, and the Einstein constraints.

Our paper is organized as follows. In section \[scaMax4D\], we first consider the paradigmatic example of gravity coupled to a massless scalar field and to a Maxwell field in 4 spacetime dimensions. The action (\[001\]) reads in this case $$\label{action1}
S[g_{\alpha \beta}, \phi, A_\gamma] =
{1 \over 2} \int \{ R - \partial_\mu \phi \, \partial^\mu \phi
- {1 \over 2} F_{\mu \nu} F^{\mu \nu} e^{\lambda \phi} \}
\sqrt{-g} \, d^4 x.$$ For this simple example, we shall explicitly determine the subcritical domain 