  In this paper, we first give a new simple proof to the elimination theorem of
definite fold by homotopy for generic smooth maps of manifolds of dimension
strictly greater than $2$ into the $2$--sphere or into the real projective
plane. Our new proof has the advantage that it is not only constructive, but is
also algorithmic: the procedures enable us to construct various explicit
examples. We also study simple stable maps of $3$--manifolds into the
$2$--sphere without definite fold. Furthermore, we prove the non-existence of
singular Legendre fibrations on $3$--manifolds, answering negatively to a
question posed in our previous paper.
