be included without changing the asymptotics of the solutions. Explicitly, we add a (nonlinear) potential term, $$\label{actionnl}
S_{\rm NL}[g_{\alpha \beta}, \phi]  =
- \int V(\phi) \sqrt{-g} \, d^D x,$$ to the action (\[001\]), where $V(\phi)$ must fulfill some assumptions given in section \[nonlinear\]. $V(\phi)$ may, for example, be an exponential function of $\phi$, a constant, or a suitable power of $\phi$. Similar forms for $V(\phi)$ were considered with $D=4$ in [@R00b].

Finally, in section \[conclusions\], we state two theorems that summarize the main results of the paper and give concluding remarks.

On the generality of our construction
-------------------------------------

As we shall see the number of arbitrary functions contained in solutions to the velocity-dominated constraint equations is equal to the number of arbitrary functions for solutions to the Einstein-matter constraints. In this function-counting sense, our construction describes what is customarily called a “general” solution of the system. Intuitively speaking, our construction concerns some “open set” of the set of all solutions (indeed, the Kasner-like behaviour of the solution is unchanged under arbitrary, small perturbations of the initial data, because this simply amounts to changing the integration functions). Note that, in the physics literature, such a “general” solution is often referred to 