  In this work, we explore the construction of the most general vector-tensor
theory with an SU(2) global symmetry in the vector sector as a proposal for a
modified theory of gravity. We start with a general Lagrangian containing terms
involving symmetric and/or antisymmetric combinations of the covariant
derivative of the vector field plus an arbitrary function of the vector field
times the Ricci scalar. Then, we study the degeneracy of the full theory to
determine whether it can be healthy or not. We find relations among some of the
free functions in the Lagrangian that are necessary for the healthiness of the
theory in correspondence with the several ways in which the kinetic matrix can
be turned degenerate. Finally, we take the decoupling limit of the theory and
find additional conditions on the free functions that are necessary for the
healthiness of the longitudinal modes.
