  We discover two additional Lax pairs and three nonlocal recursion operators
for symmetries of the general heavenly equation introduced by Doubrov and
Ferapontov. Converting the equation to a two-component form, we obtain
Lagrangian and Hamiltonian structures of the two-component general heavenly
system. We study all point symmetries of the two-component system and, using
the inverse Noether theorem in the Hamiltonian form, obtain all the integrals
of motion corresponding to each variational (Noether) symmetry. We discover
that in the two-component form we have only a single nonlocal recursion
operator. Composing the recursion operator with the first Hamiltonian operator
we obtain second Hamiltonian operator. We check the Jacobi identities for the
second Hamiltonian operator and compatibility of the two Hamiltonian structures
using P. Olver's theory of functional multi-vectors. Our well-founded
conjecture is that P. Olver's method works fine for nonlocal operators. We show
that the general heavenly equation in the two-component form is a
bi-Hamiltonian system integrable in the sense of Magri. We demonstrate how to
obtain nonlocal Hamiltonian flows generated by local Hamiltonians by using
formal adjoint recursion operator.
