Rich gauge structures from an unitary approach of some massless gauge fields of spins one and two

Considering a bosonic ($1$-)form-valued $k$-form with a second-order Lagrangian dynamics [depending on two arbitrary real constants] we firstly perform the Dirac analysis. The procedure implies a partition of cardinally seven for the plane of the real parameters that label the starting Lagrangian. In each of the seven partition's components one determines the number and the nature of independent degrees of freedom and also a generating set of gauge transformations. Secondly, with the help of some auxiliary gauge/matter tensor gauge fields, in each of the seven situations, we construct the first-order Lagrangian density corresponding to the second-order one.