  We present the first statistical analysis of kinematically-resolved,
spatially-extended Ly$\alpha$ emission around $z = 2-3$ galaxies in the Keck
Baryonic Structure Survey (KBSS) using the Keck Cosmic Web Imager (KCWI). Our
sample of 59 star-forming galaxies ($z_\mathrm{med} = 2.29$) comprises the
subset with typical KCWI integration times of $\sim 5$ hours and with existing
imaging data from the Hubble Space Telescope and/or adaptive optics-assisted
integral field spectroscopy. The high-resolution images were used to evaluate
the azimuthal dependence of the intensity distribution and kinematics of
diffuse Ly$\alpha$ emission with respect to the stellar continuum within
projected galactocentric distances of $\lesssim 30$ proper kpc. We introduce
cylindrically-projected 2D spectra (CP2D) that map the averaged Ly$\alpha$
spectral profile over a specified range of azimuthal angle, as a function of
impact parameter around galaxies. The averaged CP2D spectrum of all galaxies
shows clear signatures of Ly$\alpha$ resonant scattering by outflowing gas.
When the CP2D spectra are binned using ranges of azimuthal angle corresponding
to the projected major and minor axes, the spectra in two bins are almost
consistent with only minor residual that comprises $\le 2\%$ ($\sim 2 \sigma$)
of the integrated Ly$\alpha$ emission. The symmetry implies that the Ly$\alpha$
scattering medium is dominated by outflows in all directions within 30 kpc.
Meanwhile, we find that the blueshifted component of Ly$\alpha$ emission is
marginally stronger along galaxy minor axes for galaxies with relatively weak
Ly$\alpha$ emission. We speculate that this weak directional dependence of
Ly$\alpha$ emission becomes discernible only when the Ly$\alpha$ escape
fraction is low. These discoveries highlight the need for similar analyses in
simulations with Ly$\alpha$ radiative transfer modeling.
