  We prove a rigidity theorem for morphisms from products of open subschemes of
the projective line into solvable groups not containing a copy of $\Ga$ (for
example, wound unipotent groups). As a consequence, we deduce several
structural results about unirational group schemes, including that
unirationality for group schemes descends through separable extensions. We also
apply the main result to prove that permawound unipotent groups are unirational
and -- when wound -- commutative.
