  The present paper finds new necessary and sufficient conditions for
$6$-dimensional closed and simply-connected manifolds of certain classes to
admit special generic maps into certain Euclidean spaces.
  The class of special generic maps naturally contains Morse functions with
exactly two singular points on spheres in so-called Reeb's theorem,
characterizing spheres topologically, and canonical projections of unit
spheres. Our paper concerns variants of Reeb's theorem. Several results are
known e. g. the cases where the manifolds of the targets are the plane and some
cases where the manifolds of the domains are closed and simply-connected. Our
paper concerns $6$-dimensional versions of a result of Nishioka, determining
$5$-dimensional closed and simply-connected manifolds admitting special generic
maps into Euclidean spaces completely. Closed and simply-connected manifolds
are central geometric objects in (classical) algebraic topology and
differential topology. The $6$-dimensional case is more complicated than the
$5$-dimensional one: they are classified via explicit algebraic systems.
