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In this article, we aim to investigate some cylindrically symmetric solutions in a very well known modified theory named as    $$f(R, \phi , X)$$  f ( R , ϕ , X ) theory of gravity, where the terms  R ,    $$\phi $$  ϕ and  X  are clarified as Ricci Scalar, scalar potential, and kinetic term respectively. For this purpose, we consider the cylindrically symmetric space-time to discuss the cylindrical solutions in some realistic regions. We further discuss six distinct cases of exact solutions using the field equations of    $$f(R, \phi , X)$$  f ( R , ϕ , X ) modified theory of gravity. Furthermore, we set some suitable values of    $$U_0$$  U 0 and    $$\alpha $$  α in    $$f(R, \phi , X)=R+\alpha R^2 - V(\phi )+ X$$  f ( R , ϕ , X ) = R + α R 2 - V ( ϕ ) + X for the investigation of well-known Levi–Civita and cosmic string solutions. The Energy conditions are also investigated for all different cases and observed that null energy conditions are violated, which is the indication of the existence of cylindrical wormholes.

A study of cylindrically symmetric solutions in $$f(R, \phi , X)$$ theory of gravity