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tokens: can be noticed as a non-local equation of state as it describes the radial pressure as a function of the energy density at a specific point on the manifold (under the constraint f ( R, T, Q ) = R ). Now, we consider two subcases as follows. 4.1 The Gokhroo and Mehra Ansätz In order to continue the systematic analysis, we take the assumption introduced by Gokhroo and Mehra, as where K ∈ (0, 1), and μ o is a constant. Utilizing equation ( 59 ) in equations ( 10 ) and ( 17 ), we get and Further, from equations ( 11 ) and ( 12 ), we obtain It could be useful to introduce new structural variables as In this context, after substituting equation ( 63 ) in equation ( 62 ), we obtain which appears to be in form proposed by Ricatti. The integration of this expression produces the line element in terms of z and Π in the background of f ( R, T, Q ) theory as where C is integration constant. Thus, one may write the effective forms of physical variables as Such kind of solutions could be helpful to understand some hidden and very interesting features of the spherical systems. Di Prisco et al. ( 2011 ) calculated such results in the background of GR. These results were modified by Sharif and Yousaf for the cases of cylindrically Sharif Yousaf ( 2012b ) and shearfree spherical Sharif Yousaf ( 2012a ) structures. Yousaf ( 2017 ) and Bhatti ( 2016 ) further extended these results for Einstein-Λ gravity. To obtain non-singular characteristics of the obtained solutions, the obtained solutions ( 66 )–( 68 ) should satisfy equations ( 19 )–( 21 ) at the boundary. 4.2 The polytropic fluid with zero complexity factor In this subsection, we study polytropic relativistic fluid in the presence of f ( R, T, Q ) corrections. To deal with the system of equations, we require vanishing complexity factor condition to supplement with the polytropic equation of state. Here, we consider two cases of polytropes separately. The first one is with the polytropic constant K , polytropic exponent γ and polytropic index n . To make the dimensionless form of TOV equation and the mass function, we may define some new variables as where subscript c shows that the quantity is calculated at the centre. At the boundary r = r Σ (ξ = ξ Σ ), we have ψ(ξ Σ ) = 0. Putting these dimensionless variables in TOV equation, we get Differentiating ν(ξ) and substituting the value of m ( r ), we get In two ordinary differential equations ( 72 ) and ( 73 ), there are three unknown functions ν, ψ, and Π. For the unique solution of these equations, we need an extra condition. To obtain this, we assume the vanishing complexity factor condition in form of dimensionless variables. This gives These differential equations have a unique solution for arbitrary values of two parameters n and α. Any solution of these equations provide the mass, density, radius, and pressure of peculiar stellar object for the specified values of n and α. Now, we consider the second case of polytropic equation of state , where μ b