  We present and analyze three-dimensional data cubes of Neptune from the
OSIRIS integral-field spectrograph on the 10-m Keck telescope, from July 2009.
These data have a spatial resolution of 0.035"/pixel and spectral resolution of
R~3800 in the H and K broad bands. We focus our analysis on regions of
Neptune's atmosphere that are near-infrared dark- that is, free of discrete
bright cloud features. We use a forward model coupled to a Markov chain Monte
Carlo algorithm to retrieve properties of Neptune's aerosol structure and
methane profile above ~4 bar in these near-infrared dark regions.
  Using a set of high signal-to-noise spectra in a cloud-free band from 2-12N,
we find that Neptune's cloud opacity is dominated by a compact, optically thick
cloud layer with a base near 3 bar and composed of low albedo, forward
scattering particles, with an assumed characteristic size of ~1$\mu$m. Above
this cloud, we require a vertically extended haze of smaller (~0.1 $\mu$m)
particles, which reaches from the upper troposphere (~0.6 bar) into the
stratosphere. The particles in this haze are brighter and more isotropically
scattering than those in the deep cloud. When we extend our analysis to 18
cloud-free locations from 20N to 87S, we observe that the optical depth in
aerosols above 0.5 bar decreases by a factor of 2-3 or more at mid- and
high-southern latitudes relative to low latitudes.
  We also consider Neptune's methane (CH$_4$) profile, and find that our
retrievals indicate a strong preference for a low methane relative humidity at
pressures where methane is expected to condense. Our preferred solution at most
locations is for a methane relative humidity below 10% near the tropopause in
addition to methane depletion down to 2.0-2.5 bar. We tentatively identify a
trend of lower CH$_4$ columns above 2.5 bar at mid- and high-southern latitudes
over low latitudes.
