  We investigate the chiral phase structure of quark matter with spheroidal
momentum-space anisotropy specified by one anisotropy parameter $\xi$ in the
2+1 flavor quark-meson model. We find that the chiral phase diagram and the
location of the critical endpoint (CEP) are affected significantly by the value
of $\xi$. With the increase of $\xi$, the CEP is shifted to smaller
temperatures and larger quark chemical potentials. And the temperature of the
CEP is more sensitive to the anisotropy parameter than the corresponding quark
chemical potential, which is opposite to the study for finite system volume
effect. Furthermore, the effects of momentum anisotropy on the thermodynamic
properties and scalar (pseudoscalar) meson masses are also studied at vanishing
quark chemical potential. The numerical results show that an increase of $\xi$
can hinder the restoration of chiral symmetry. We also find that shear
viscosity and electrical conductivity decrease as $\xi$ grows. However, bulk
viscosity exhibits a significant non-trivial behavior with $\xi$ in the entire
temperature domain of interest.
