We investigate two Type-IIa Minimally Modified Gravity theories, namely VCDM
and Cuscuton theories. We confirm that all acceptable Cuscuton solutions are
always solutions for VCDM theory. However, the inverse does not hold. We find
that VCDM allows for the existence of exact General Relativity (GR) solutions
with or without the presence of matter fields and a cosmological constant. We
determine the conditions of existence for such GR-VCDM solutions in terms of
the trace of the extrinsic curvature and on the fields which define the VCDM
theory. On the other hand, for the Cuscuton theory, we find that the same set
of exact GR solutions (such as Schwarzschild and Kerr spacetimes) is not
compatible with timelike configurations of the Cuscuton field and therefore
cannot be considered as acceptable solutions. Nonetheless, in Cuscuton theory,
there could exist solutions which are not the same but close enough to GR
solutions. We also show the conditions to determine intrinsic-VCDM solutions,
i.e. solutions which differ from GR and do not belong to the Cuscuton model. We
finally show that in cosmology a mapping between VCDM and the Cuscuton is
possible, for a generic form of the VCDM potential. In particular, we find that
for a quadratic potential in VCDM theory, this mapping is well defined giving
an effective redefinition of the Planck mass for the cosmological background
solutions of both theories.
