  In this paper we construct a family of cohomology classes on the moduli space
of stable curves generalizing Witten's $r$-spin classes. They are parameterized
by a phase space which has one extra dimension and in genus $0$ they correspond
to the extended $r$-spin classes appearing in the computation of intersection
numbers on the moduli space of open Riemann surfaces, while when restricted to
the usual smaller phase space, they give in all genera the product of the top
Hodge class by the $r$-spin class. They do not form a cohomological field
theory, but a more general object which we call F-CohFT, since in genus $0$ it
corresponds to a flat F-manifold. For $r=2$ we prove that the partition
function of such F-CohFT gives a solution of the discrete KdV hierarchy.
Moreover the same integrable system also appears as its double ramification
hierarchy.
