  The well known N=2 string theory describes self-dual gravity, as was shown by
Ooguri and Vafa sometime ago. In search of a variant of this theory which would
describe self-dual supergravity in 2+2 dimensions, we have constructed two new
N=2 strings theories in which the target space is a superspace. Both theories
contain massless scalar and spinor fields in their spectrum, and one of them
has spacetime supersymmetry. However, we find that the interactions of these
fields do not correspond to those of self-dual supergravity. In our
construction, we have used the basic (2,2) superspace variables, and considered
quadratic constraints in these variables. A more general construction may be
needed for a stringy description of self-dual supergravity.
