New Exact Spherically Symmetric Solutions in $f(R,\phi,X)$ gravity by
  Noether's symmetry approach
  The exact solutions of spherically symmetric space-times are explored by
using Noether symmetries in $f(R,\phi,X)$ gravity with $R$ the scalar
curvature, $\phi$ a scalar field and $X$ the kinetic term of $\phi$. Some of
these solutions can represent new black holes solutions in this extended theory
of gravity. The classical Noether approach is particularly applied to acquire
the Noether symmetry in $f(R,\phi,X)$ gravity. Under the classical Noether
theorem, it is shown that the Noether symmetry in $f(R,\phi,X)$ gravity yields
the solvable first integral of motion. With the conservation relation obtained
from the Noether symmetry, the exact solutions for the field equations can be
found. The most important result in this paper is that, without assuming
$R=\textrm{constant}$, we have found new spherically symmetric solutions in
different theories such as: power-law $f(R)=f_0 R^n$ gravity, non-minimally
coupling models between the scalar field and the Ricci scalar $f(R,\phi,X)=f_0
R^n \phi^m+f_1 X^q-V(\phi)$, non-minimally couplings between the scalar field
and a kinetic term $f(R,\phi,X)=f_0 R^n +f_1\phi^mX^q$ , and also in extended
Brans-Dicke gravity $f(R,\phi,X)=U(\phi,X)R$. It is also demonstrated that the
approach with Noether symmetries can be regarded as a selection rule to
determine the potential $V(\phi)$ for $\phi$, included in some class of the
theories of $f(R,\phi,X)$ gravity.
