Let omega be a bi-form of degree (r, s) on J^k(R^n, R^m).  Then omega is called dH closed if dH(omega) = 0 where dH denotes the horizontal exterior derivative and dH exact if there exists a bi-form eta of degree (r - 1, s) on J^l(R^n, R^m) such that omega = dH(eta).  If r < n, then every dH closed form is  dH exact.  If r = n, s > 0 and the integration by parts operator applied to omega yields I(omega) = 0, then omega is dH exact.  If r = n, s = 0 and the EulerLagrange operator applied to omega yields E(omega) = 0, then omega is dH exact.

If dH(omega) = 0 or I(omega) = 0 then there are numerous algorithms for finding a bi-form eta such that omega = dH(eta).  One approach is to use the horizontal homotopy operators.  These operators are defined in terms of the higher Euler operators--the precise formulas are quite complex and the user is referred to Ian M. Anderson, Notes on the Variational Bicomplex

For s > 0 these operators are total differential operators and therefore, unlike the usual homotopy operators for the deRham complex or the vertical homotopy operators for bi-forms on jet spaces, do not involve any quadratures.  For s = 0 the horizontal homotopy does involve quadratures and the optional arguments used in the commands DeRhamHomotopy or VerticalHomotopy can be invoked.

If omega is a bi-form of degree (r, s) with r > 0, then HorizontalHomotopy(omega) returns a bi-form of degree (r - 1, s).

The command HorizontalHomotopy is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form HorizontalHomotopy(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-HorizontalHomotopy(...).