Vortex Lattice Anisotropy in Sr$_2$RuO$_4$ with $H \parallel a$*

We have studied the vortex lattice (VL) in superconducting Sr$_2$RuO$_4$ using spin-polarized small-angle neutron scattering (SP-SANS) and with magnetic fields ($H$) close to, but not perfectly aligned with, the crystalline basal plane. In this configuration the VL possesses a large transverse field component, due to the large anisotropy of Sr$_2$RuO$_4$, which greatly increases the spin-flip SANS scattered intensity. Bragg reflections indicative of a highly distorted hexagonal VL were observed, and an analysis of the magnitude of the scattering vector yielded an anisotropy of $50 - 60$, roughly two times the $H_{c2}$ anisotropy. We discuss implications of this result in relation to theoretical predictions of Pauli paramagnetic effects in Sr$_2$RuO$_4$ with $H \perp c$ [K. Machida and M. Ichioka, Phys. Rev. B {\bf 77}, 184515 (2008)].