Exact black hole solutions in shift symmetric scalar-tensor theories

We derive a variety of exact black hole solutions in a subclass of Horndeski's scalar-tensor theory possessing shift symmetry, phi -> phi + c and reflection symmetry, phi -> -phi The theory admits two arbitrary functions of X := -(partial derivative phi)(2)/2, and our solutions are constructed without specifying the concrete form of the two functions, implying that black hole solutions in specific scalar-tensor theories found in the literature can be extended to a more general class of theories with shift symmetry. Our solutions include a black hole in the presence of an effective cosmological constant, the Nariai spacetime, the Lifshitz black hole, and other nontrivial solutions, all of which exhibit nonconstant scalar-field profiles.