Comparing metric and Palatini approaches to vector Horndeski theory

We compare cosmologic and spherically symmetric solutions to metric and Palatini versions of vector Horndeski theory. It appears that Palatini formulation of the theory admits more degrees of freedom. Specifically, homogeneous isotropic configuration is effectively bimetric, and static spherically symmetric configuration contains non-metric connection. In general, the exact solution in metric case coincides with the approximative solution in Palatini case. The Palatini version of the theory appears to be more complicated, but the resulting non-linearity may be useful: we demonstrate that it allows the specific cosmological solution to pass through singularity, which is not possible in metric approach.