  We construct all (2+1)-dimensional PDEs depending only on 2nd-order
derivatives of unknown which have the Euler-Lagrange form and determine the
corresponding Lagrangians. We convert these equations and their Lagrangians to
two-component forms and find Hamiltonian representations of all these systems
using Dirac's theory of constraints. We consider three-parameter integrable
equations that are cubic in partial derivatives of the unknown applying our
method of skew factorization of the symmetry condition. Lax pairs and recursion
relations for symmetries are determined both for one-component and
two-component forms. For cubic three-parameter equations in the two-component
form we obtain recursion operators in $2\times 2$ matrix form and
bi-Hamiltonian representations, thus discovering three new bi-Hamiltonian (2+1)
systems.
