2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153831Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced which is called ``universal" since it plays an important role for all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit formulas are presented, in all the others recurrence relations are given which enable one to obtain the identities.LaTeX, 11 pagesExactly Solvable and Integrable Systemstext