Figs. 4 and 5. The central energy density increases as we move along these curves from the right hand side. The portion of the curves left to the solid circles represent the equilibrium sequences of hybrid stars with CFL quark matter core. The curves between the solid circle and triangle represent the sequences of hybrid star composed of 2SC quark matter core. It is clear from the lower panel of Fig. 4 and upper panel of Fig. 5 that the stable configuratio ns of hybrid stars with CFL quark matter core belong to third family of compact stars. Further, irrespective of the choice of the EOS of the nuclear matter, the stable configurations of the non-r otating hybrid stars exist within the NJL model only when the EOSs for the CSQM are constructed for G$_{D}$ = 1 . 2 G$_{S}$ with G$_{V}$ ≲ 0 . 1 G$_{S}$ . These values of G$_{D}$ and G$_{V}$ , for which the stable configurations of the hybrid star exist , are very much similar to the ones found in Ref. [33]. It appears that the stability of the hybrid stars with CSQM core depends solely on the choice of the EOS for the CSQM. However, the composition of the hybrid stars depend on the behaviour of the nuclear matter EOS. For instance, in case of the TM1 and the NL3 EOSs of the nuclear matter, the core of the hybrid stars are composed of CSQM which is either in 2SC phase or in the CFL phase. In the later case, CFL quark matter core is surrounded by a layer of 2SC quark matter with the outer layer composed of nuclear matter . The thickness of the 2SC quark matter at the maximum hybrid star mass is around 0 . $^{5}$−0 . 7km and its mass is ∼ 0 . 1 $^{M}$⊙ . On the other hand, no 2SC quark matter appears in the stable configurations of the hybrid star constructed using the EOSs for which the nuclear matter part correspond to the APR and SLY4. The equilibrium sequences for the hybrid stars rotating with fixed rotation frequency f are constructed. As an illustration, in Fig. 6, we plot mass verses circumferential equatorial radius R$_{eq}$ at fixed values of the rotational frequency obtained for two d i ff erent EOSs. For the clarity, we mainly focus on the regions of the M − R$_{eq}$ curves corresponding to the sequences of the hybrid stars which are relevant in the present context. We see that beyond certain frequency, so-called the critical rotation frequency f$_{crit}$ , the stable configuration for the rotating hybrid star does n ot exist. The solid black lines in Fig. 6 represent the result obtained at the f = f$_{crit}$ . In Fig. 7 we plot the values for the f$_{crit}$ (left panel) calculated for the cases for which the stable configurations for the non-rotating hybrid star exist. It is evident that the values of f$_{crit}$ are quite sensitive to the choice of the EOS for the nuclear matter as well as the CSQM. Depending on the EOSs, f$_{crit}$ varies in the range of 350 − 1275 Hz. We also plot in Fig. 7 (right panel) the maximum mass M$_{max}$ for the non-rotating hybrid stars with CSQM core. It is interesting to note from this Fig. that the values of G$_{D}$ and G$_{V}$ for a given nuclear matter EOS can be so adjusted that the resulting hybrid 7