Geymonat extended Gurtin’s result on Beltrami’s completeness to [Formula: see text]-case. The first objective of this paper is to generalize Geymonat’s result to [Formula: see text]-case with tangential and normal boundary conditions. Maggiani et al. proposed an extension of Beltrami’s-type decomposition for symmetric matrix fields in [Formula: see text] when [Formula: see text] is simply connected domain and of class [Formula: see text]. The second objective of this paper is to present more extensions of the above decomposition when the domain [Formula: see text] is only of class [Formula: see text] and not necessarily simply connected. The third objective of this paper is to use the previous characterizations of matrix fields to solve some variants of the bi-Laplacian problem.