  A one-parameter generalization of the hierarchy of negative flows is
introduced for integrable hierarchies of evolution equations, which yields a
wider (new) class of non-evolutionary integrable nonlinear wave equations. As
main results, several integrability properties of these generalized negative
flow equation are established, including their symmetry structure, conservation
laws, and bi-Hamiltonian formulation. (The results also apply to the hierarchy
of ordinary negative flows). The first generalized negative flow equation is
worked out explicitly for each of the following integrable equations: Burgers,
Korteweg-de Vries, modified Korteweg-de Vries, Sawada-Kotera, Kaup-Kupershmidt,
Kupershmidt.
