New Exact Spherically Symmetric Solutions in $f(R,\phi,X)$ gravity by Noether's symmetry approach

Understand

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.