The author recalls the well-known properties of the homotopy operator \(H\) in the module of all exterior differential forms on a starshaped subset of a manifold; together with the common exact forms \(\omega\) (with \(d\omega= 0\)) also the antiexact forms (satisfying \(H\omega= 0\)) are mentioned. Then the antiexact forms and the homotopy operator are used to resolve the system \[ d\Omega+ \Gamma\wedge \Omega= \Sigma,\;d\Sigma+ \Gamma\wedge \Sigma= \Theta,\;d\Gamma+ \Gamma\wedge \Gamma= \Theta,\;d\Theta+ \Gamma\wedge \Theta= \Theta\wedge \Gamma \] by quadratures (the proof is unfortunately only referred to). Here \(\Omega\), \(\Sigma\) are column matrices of \(k\)-forms or \((k+1)\)-forms, respectively, and \(\Gamma\), \(\Theta\) are square matrices of 1-forms or 2-forms, respectively. This is the so-called unconstrained problem, however, if certain additional conditions of \(\Gamma\) (the connections) or \(\Theta\) (the curvatures) are supposed, then the solution can also be obtained. The results are applied to several classical examples, e.g., to the determination of various contact structures and to a generalization of the Frobenius theorem.