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We consider the possibility for the existence of the third family of compactobjects, considering the effect of strong magnetic fields inside the hybridstars. As a result, we demonstrate such new sequences of stable equilibriumconfigurations for some hadronic equations of state. Through the analysis ofthe adiabatic index inside stars, we find the conditions for appearing thethird family of compact objects, i.e., for hadronic stars without quarks, thatthe maximum mass should be small, the central density for the maximum massshould be also small, and the radius for the the maximum mass should be large.Even for soft hadronic equations of state, the two solar-mass stars mightsurvive as the third family of compact objects, once quark matter with strongmagnetic field, such as $\sim {\cal O}(10^{19} {\rm G})$, is taken intoaccount. It might give a hint to solve the so-called hyperon puzzle in nuclearphysics.

Quark matter with strong magnetic field and possibility of the third family of compact stars