  Superstrings have been postulated based on parafermionic partition functions
which permit spacetime supersymmetry by generalized Jacobi identities. A
comprehensive search finds new such identities. Quadrilateral anomaly
cancellation gives constraints on allowed chiral fermions. Bosonic left-movers
and $Z_4$ parafermionic right-movers combine in a new heterotic superstring,
more constrained than the old one, yet equally applicable to physics.
