  We have studied the properties of hot and dense quark matter based on the
3-flavor Nambu-Jona-Lasinio (NJL) model as well as its Polyakov-loop extension
(pNJL) with scalar-isovector and vector-isovector couplings. Provided a
considerable large isospin asymmetry or isospin chemical potential, isospin
splittings of constituent mass, chiral phase transition boundary, and critical
point for $u$ and $d$ quarks can be observed for positive isovector coupling
constants but are suppressed for negative ones. The quark matter symmetry
energy decreases with the increasing isovector coupling constant, and is mostly
enhanced in the pNJL model than in the NJL model. A positive scalar-isovector
coupling constant is more likely to lead to an unstable isospin asymmetric
quark matter. The isovector coupling has been further found to affect particle
fractions as well as the equation of state in hybrid stars. Possible effects on
the isospin properties of quark matter have also been discussed if the
strangeness sector is further broken among the flavor symmetry.
