  We investigate the static and spherically symmetric solutions in a class of
the generalized Proca theory with the nonminimal coupling to the Einstein
tensor. First, we show that the solutions in the scalar-tensor theory with the
nonminimal derivative coupling to the Einstein tensor can be those in the
generalized Proca theory with the vanishing field strength. We then show that
when the field strength takes the nonzero value the static and spherically
symmetric solutions can be found only for the specific value of the nonminimal
coupling constant. Second, we investigate the first-order slow-rotation
corrections to the static and spherically symmetric background. We find that
for the background with the vanishing electric field strength the slowly
rotating solution is identical to the Kerr- (anti-) de Sitter solutions in
general relativity. On the other hand, for the background with the nonvanishing
electric field strength the stealth property can realized at the first order in
the slow-rotation approximation.
