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A second-order generalization of the fundamental Lepage form of geometriccalculus of variations over fibered manifolds with 2-dimensional base isdescribed by means of insisting on (i) equivalence relation "Lepagedifferential 2-form is closed if and only if the associated Lagrangian istrivial" and, (ii) the principal component of Lepage form, extending thewell-known Poincar\'{e}-Cartan form, preserves order prescribed by a givenLagrangian. This approach completes several attempts of finding a Lepageequivalent of a second-order Lagrangian possessing condition (i), which iswell-known for first-order Lagrangians in field theory due to Krupka andBetounes.

The fundamental Lepage form in two independent variables: a generalization using order-reducibility