  Linear and nonlinear Hodge-like systems for 1-forms are studied, with an
assumption equivalent to complete integrability substituted for the requirement
of closure under exterior differentiation. The systems are placed in a
variational context and properties of critical points investigated. Certain
standard choices of energy density are related by Backlund transformations
which employ basic properties of the Hodge involution. These Hodge-Backlund
transformations yield invariant forms of classical Backlund transformations
that arise in diverse contexts. Some extensions to higher-degree forms are
indicated.
