Flat pencils of symplectic connections and Hamiltonian operators of degree 2

Bi-Hamiltonian structures involving Hamiltonian operators of degree 2 are studied. Firstly, pairs of degree 2 operators are considered in terms of an algebra structure on the space of 1-forms, related to so-called Fermionic Novikov algebras. Then, degree 2 operators are considered as deformations of hydrodynamic type Poisson brackets.