Geometric integration by parts and Lepage equivalents

We compare the integration by parts of contact forms - leading to definition interior Euler operator with so-called canonical splittings variational morphisms. In particular, we discuss possibility a generalization first method lower degree. define suitable Residual for this case and, working out an original conjecture Olga Rossi, recover Krupka-Betounes equivalent order field theories. A second is discussed.