Spherically symmetric distributions with an invariant and vanishing complexity factor by means of the extended geometric deformation

In this work, we will analyze the complexity factor, proposed by L. Herrera, of spherically symmetric static distribution through the gravitational decoupling method. Specifically, we will consider both spatial and temporal deformations of the metric function, and we will impose conditions over the complexity factor to close the system of equations. In particular, we found that the regularity at the center of both the seed and final solutions led to important restrictions on the deformation of the spatial metric components. These are particularly restrictive for the MGD method. In this case, we show that if the seed solution is regular at r=0, the final solution with invariant complexity factor will be singular at this point unless f=0. We also show that solutions with the same temporal components will, in general, lead to the same solutions with vanishing complexity factor in the MGD approach. Finally, we will construct realistic models using different seed solutions such as Tolman IV and FS (Finch-Skeas).