  A class of hybrid (topologically) massive off-shell supergravities coupled to
an on-shell matter scalar multiplet was recently constructed. The auxiliary
field in the off-shell multiplet is dynamical for generic values of the eight
parameters. We find that by choosing the parameters appropriately, it remains
non-dynamical. We perform linearized analysis around the supersymmetric AdS3
vacuum and its Minkowski limit. The ghost-free condition for the Minkowski
vacuum is explored. For the AdS3 vacuum, we obtain the criticality condition
and find that at the critical points, one of the two massive gravitons becomes
pure gauge and decouples from the bulk physics, whilst the other has positive
energy. We demonstrate that the mass of the BTZ black hole is non-negative at
the critical points. We also investigate general BPS solutions. For certain
parameter choices, we obtain exact solutions. In particular, we present the BPS
string (domain-wall) solution that is dual to certain two-dimensional quantum
field theory with an ultra-violet conformal fixed point.
