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We study point symmetries, the corresponding conserved densities andhierarchies of four new bi-Hamiltonian heavenly systems in 3+1 dimensions whichwe discovered recently. We exhibit an important role played by the inverserecursion operators in the description of the hierarchies. Their use istwofold, either to provide the correct bi-Hamiltonian representation or togenerate nonlocal symmetry flows. Invariant solutions w.r.t. nonlocalsymmetries will generate (anti-)self-dual gravitational metrics which do notadmit Killing vectors which is a characteristic feature of $K3$ gravitationalinstanton.

Symmetries, integrals and hierarchies of new (3+1)-dimensional bi-Hamiltonian systems of Monge--Amp\`ere type