  Solutions with degenerate metric ($det(g_{\mu\nu})=0$ or $g_{\mu\nu}=0$) in
the first order formalism (FOF) are physically acceptable: they may describe
topology changes (Horowitz) and reduction of "metrical dimension" (Tseytlin) of
space-time. The latter implies disappearance of the volume element
$\sqrt{-g}d^4x$ of 4-D space-time. We pay attention that besides $\sqrt{-g}$,
the 4-D space-time differentiable manifold possesses also a "manifold volume
measure" (MVM)described by a 4-form which is sign indefinite and generically
independent of the metric. The FOF proceeds with originally independent
connection and metric structures of the space-time manifold. We bring up the
question whether the FOF should be supplemented with degrees of freedom of MVM.
Adding such manifold degrees of freedom to the action principle in the FOF we
realize very interesting dynamics. Such Two Measures Theory enables radically
new approaches to resolution of the cosmological constant problem. We show that
fine tuning free solutions describing a transition to $\Lambda =0$ state
involve oscillations of $g_{\mu\nu}$ and MVM around zero. The latter can be
treated as a dynamics involving changes of orientation of the space-time
manifold. As we have shown earlier, in realistic scale invariant models (SIM),
solutions formulated in the Einstein frame satisfy all existing tests of
General Relativity (GR). Here we reveal surprisingly that in SIM, all ground
state solutions with $\Lambda\neq 0$ appear to be degenerate either in $g_{00}$
or in MVM. Sign indefiniteness of MVM in a natural way yields a dynamical
realization of a phantom cosmology ($w<-1$). For all solutions, the metric
tensor rewritten in the Einstein frame has regularity properties exactly as in
GR. A possibility of a strong gravity effect in LHC experiments is discussed.
