^q =
\bar{v}^q$ (where the Ricci scalar (\[ric2\]) diverges) for all $q < 0$ or $q > 0$ (implying $q
> 1/2$). For $q < 0$, the Penrose diagram is a triangle bounded by past null infinities $\bar{u} \to
-\infty$ and $\bar{v} = 0$ (which is at infinite geodesic distance). For $q
> 0$, geodesics terminate at $\bar{v} = 0$, unless $q = n$ integer. For $n$ even, the extended spacetime has two central curvature singularities $r = 0$, one spacelike and the other timelike. The extended spacetime for $n$ odd is more realistic. In this case the extension from $\bar{v}
> 0$ to $\bar{v} < 0$ amounts to replacing (\[gar3\]) with $A > 0$ by the original Garfinkle solution (\[gar1\]) with $A > 0$, the resulting Penrose diagram being that of Fig. 2.

A new CSS solution for $\L = 0$
===============================

Among the one-parameter ($c$ or $q$) family of CSS solutions (\[gar1\]), the special solution, corresponding to $\k c^2 = 1$, ds\^2 = A( + )\^4 - 14(v+u)\^2d\^2, is singled out by the fact that the transformation (\[q\]) breaks down for this value. The transformation appropriate to this case, -u = 2e\^[-U]{}, v = 2e\^V = 2e\^[U-2T]{} (with $T \ge U$ for $u+v \le 0$) transforms the solution (\[gar4\]) to ds\^2 & = & e\^[-2U]{}\[-4A(1 + e\^[U-T]{})\^4dUdV - (1 - e\^[2(U-T)]{})\^2 d\^2\],\
& = & U - 2(1 + e\^[U-T]{}) (we use from now on units such that $\k = 1$, and have dropped an irrelevant additive constant from $\phi$).

Starting from this special CSS solution of the Garfinkle class, we now derive, by a limiting process, a new CSS solution which, as we shall see, exhibits a null singularity. We translate $T$ to $T-T_0$, and take the late-time limit $T_0 \to
-\infty$, leading to the new CSS solution (written for $A = -1/2$) ds\^2 = e\^[-2U]{}(2dUdV - d\^2), = U, with a very simple form which is reminiscent of the Hayward critical solution for scalar field collapse in 3+1 dimensions [@hay], ds\^2 = e\^[2]{}(2d\^2 - 2d\^2 - d\^2), = . The transformation |[u]{} = -e\^[-2U]{}, |[v]{} = V leads from (\[crit0\]) to the even 