  In this paper we investigate the algebraic structure related to a new type of
correlator associated to the moduli spaces of $S^1$-parametrized curves in
contact homology and rational symplectic field theory. Such correlators are the
natural generalization of the non-equivariant linearized contact homology
differential (after Bourgeois-Oancea) and give rise to an invariant Nijenhuis
(or hereditary) operator (\`a la Magri-Fuchssteiner) in contact homology which
recovers the descendant theory from the primaries. We also sketch how such
structure generalizes to the full SFT Poisson homology algebra to a (graded
symmetric) bivector. The descendant hamiltonians satisfy to recursion
relations, analogous to bihamiltonian recursion, with respect to the pair
formed by the natural Poisson structure in SFT and such bivector. In case the
target manifold is the product stable Hamiltonian structure $S^1\times M$, with
$M$ a symplectic manifold, the recursion coincides with genus $0$ topological
recursion relations in the Gromov-Witten theory of $M$.
