  We develop a systematic study of the equations of motion in the first order
gravity with matter fields for degenerate metrics. Like the Hilbert-Palatini
action functional for pure gravity, the action functionals for matter fields
used are first order. These are defined for both invertible and non- invertible
metrics. Description for invertible metrics is equivalent to second order
gravity theory with matter. For degenerate metrics the theory describes a
different phase. The analysis for tetrads with one zero eigen value in theory
with scalar, Abelian vector gauge and fermion matter fields is presented in
detail.
