"Metric-affine-like" generalization of YM (mal-YM): detailed classical consideration

We consider the ``metric-affine-like'' generalization of the Yang-Mills theory (mal-YM) which we first proposed earlier. In this model, the connection is no longer assumed to be compatible with the Hermitian form in the fibers. As a consequence, along with the usual YM potential $\boldsymbol{A}_a$ and the field strength tensor $\boldsymbol{F}_{ab}$, it contains non-trivially interacting fields $\boldsymbol{B}_a$, $\boldsymbol{h}$, and $\boldsymbol{G}_{ab}$, $\boldsymbol{N}_a$, forming a non-Abelian generalization of St\"{u}ckelberg theory. Due to the spontaneous symmetry breaking $GL(n,\mathbb{C}) \to U(n)$, these new fields can be made massive and the limit $M\to\infty$ restores the standard YM theory. We perform a detailed analysis of this theory on the classical level. We discuss in detail geometric motivation for the model, field transformations, gauge symmetry and its spontaneous breaking, action, equations of motion, Noether identities, gauge fixing, and other issues.