arXiv:solv-int/9806001v1 29 May 1998 Singular sector of the KP hierarchy, ¯ ∂ -operators of non-zero index and associated integrable systems Boris G. Konopelchenko Dipartimento di Fisica, Universit´a di Lecce 73100 Lecce, Italy Luis Mart´ınez Alonso Departamento de F´ısica Te´orica, Universidad Complutense E28040, Madrid, Spain Elena Medina Departamento de Matem´aticas, Universidad de C´adiz E11510 C´adiz, Spain Abstract Integrable hierarchies associated with the singular sector of the KP hierarchy, or equivalently, with ¯ ∂ -operators of non-zero index are studied. They arise as the restriction of the standard KP hierarchy to submanifols of finite codimension in the space of independent vari- ables. For higher ¯ ∂ -index these hierarchies represent themselves fami- lies of multidimensional equations with multidimensional constraints. The ¯ ∂ -dressing method is used to construct these hierarchies. Hidden KdV, Boussinesq and hidden Gelfand-Dikii hierarchies are considered too. 1