Reconstructed black hole solutions in the scalar-tensor theory with nonminimal coupling

We consider the scalar-tensor theory witn non-minimal coupling in the Jordan frame. The action of the model contains a potential term $U(\phi )$, a coupling function $f(\phi )$. We explore a reconstruction procedure for a generic static spherically symmetric metric written in the Buchdahl parametrization: $d{s}^2={(A(u))}^{-1}d{u}^2-A(u)d{t}^2+C(u)d{\mathrm{\Omega }}^2$, with given $A(u)>0$ and $C(u)>0$. The procedure gives the relations for $U(\phi (u))$, $f(\phi (u))$ and $d\phi \mathrm{/}du$, which lead to exact solutions to equations of motion with a given metric. A key role in this approach is played by the solutions to a first order linear differential equation for the function $f(\phi (u))$. The formalism is illustrated by two examples when: a) the Reissner-Nordström-(Anti-)de Sitter metric and b) the Bocharova-Bronnikov-Melnikov-Bekenstein-(Anti)de-Sitter metric are chosen as a starting point.