On the gauge-natural operators similar to the twisted Dorfman-Courant bracket

All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE T^*E)\to T^*E)\] transforming pairs of sections \(TE T^*E \to E\) \( TE are completely described. The complete descriptions is given all generalized twisted Dorfman-Courant brackets (i.e. as above such that \(C_0\) the bracket) satisfying Jacobi identity for closed \(H\). An interesting natural characterization (usual) bracket presented.