In this work we propose a novel approach to integrate the Lane–Emden equations for relativistic anisotropic polytropes. We take advantage of the fact that Gravitational Decoupling allows to decrease the number of degrees of freedom once a well known solution of the Einstein field equations is provided as a seed so after demanding the polytropic equation for the radial pressure the system is automatically closed. The approach not only allows to extend both isotropic or anisotropic known solutions but simplifies the computation of the Tolman mass whenever the Minimal Geometric Deformation is considered given that the $$g_{tt}$$ g tt component of the metric remains unchanged. We illustrate how the the method works by analyzing the solutions obtained from Tolman IV, Durgapal IV and Wymann IIa isotropic systems as a seed for the integration.

In this work, we use the gravitational decoupling framework through the extended minimal geometric deformation to construct and study a new interior isotropic extension of Einstein’s universe solution. The resulting model fulfills the fundamental physical acceptance conditions. As well, we study the energy exchange between the Einstein’s fluid distribution and an extra perfect fluid which supports the resulting isotropic stellar configuration.

In this work, we construct a new stellar model in the regime of anisotropic fluid pressure using the concept of vanishing complexity for spherically symmetric fluid distributions (Herrera in Phys Rev D 97:044010, 2018) and a convenient ansatz in order to close the Einstein’s field equations. The resulting model fulfills the fundamental physical acceptability stellar conditions for a specific set of compactness factor. The stability and its response against fluctuations in the matter sector is also investigated.