  Integrable hierarchies associated with the singular sector of the KP
hierarchy, or equivalently, with $\dbar$-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 variables. For
higher $\dbar$-index these hierarchies represent themselves families of
multidimensional equations with multidimensional constraints. The
$\dbar$-dressing method is used to construct these hierarchies. Hidden KdV,
Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.
