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We consider (3+1)-dimensional second-order evolutionary PDEs where theunknown $u$ enters only in the form of the 2nd-order partial derivatives. Forsuch equations which possess a Lagrangian, we show that all of them have asymplectic Monge--Amp\`ere form and determine their Lagrangians. We develop acalculus for transforming the symmetry condition to a "skew-factorized" formfrom which we immediately extract Lax pairs and recursion relations forsymmetries, thus showing that all such equations are integrable in thetraditional sense. We convert these equations together with their Lagrangiansto a two-component form and obtain recursion operators in a $2\times 2$ matrixform. We transform our equations from Lagrangian to Hamiltonian form by usingthe Dirac's theory of constraints. Composing recursion operators with theHamiltonian operators we obtain the second Hamiltonian form of our systems,thus showing that they are bi-Hamiltonian systems integrable in the sense ofMagri. By this approach, we obtain five new bi-Hamiltonian multi-parametersystems in (3+1) dimensions.

Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations