In this work, we characterize a family of linear vector fields in \(\mathbb {R}^3\) that is completely integrable in the Darboux sense, thus possessing a continuum of closed trajectories. Also, we explicitly exhibit the first-order averaging functions to investigate the number of limit cycles that can bifurcate from a polynomial perturbation of degree m of the linear system contained in this family that possesses cylinders and planes as first integrals, providing isochronous centers in its intersection.
