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We investigate the mixed state properties in a type II multibandsuperconductor with uniaxial anisotropy under the Pauli paramagnetic effects.Eilenberger theory extended to a multiband superconductor is utilized todescribe the detailed vortex lattice properties, such as the flux line formfactors, the vortex lattice anisotropy and magnetic torques. We apply thistheory to Sr$_2$RuO$_4$ to analyze those physical quantities obtainedexperimentally, focusing on the interplay between the strong two-dimensionalanisotropy and the Pauli paramagnetic effects. This study allows us tounderstand the origin of the disparity between the vortex lattice anisotropy($\sim$60) and the $H_{\rm c2}$ anisotropy ($\sim$20). Among the three bands;$\gamma$ with the effective mass anisotropy $\sim$180, $\alpha$ with $\sim$120,and $\beta$ with $\sim$60, the last one is found to be the major band,responsible for various magnetic responses while the minor $\gamma$ band playsan important role in the vortex formation. Namely, in a field orientationslightly tilted away from the two dimensional basal plane those two bandscooperatively form the optimal vortex anisotropy which exceeds that given bythe effective mass formula with infinite anisotropy. This is observed by smallangle neutron scattering experiments on Sr$_2$RuO$_4$. The pairing symmetry ofSr$_2$RuO$_4$ realized is either spin singlet or spin triplet with the d-vectorstrongly locked in the basal plane. The gap structure is that the major $\beta$band has a full gap and the minor $\gamma$ band has a $d_{x^2-y^2}$ like gap.

Interplay between effective mass anisotropy and Pauli paramagnetic effects in a multiband superconductor--Application to Sr2RuO4--