  The prospect of combining integral field spectroscopy with the solar
gravitational lens (SGL) to spectrally and spatially resolve the surfaces and
atmospheres of extrasolar planets is investigated. The properties of hyperbolic
orbits visiting the focal region of the SGL are calculated analytically,
demonstrating trade offs between departure velocity and time of arrival, as
well as gravity assist maneuvers and heliocentric angular velocity. Numerical
integration of the solar barycentric motion demonstrates that navigational
acceleration of $\textrm{d}v \lesssim 80 \frac{\textrm{m}}{\textrm{s}} + 6.7
\frac{\textrm{m}}{\textrm{s}} \frac{t}{\textrm{year}}$ is needed to obtain and
maintain alignment. Obtaining target ephemerides of sufficient precision is an
open problem. The optical properties of an oblate gravitational lens are
reviewed, including calculations of the magnification and the point-spread
function that forms inside a telescope. Image formation for extended,
incoherent sources is discussed when the projected image is smaller than,
approximately equal to, and larger than the critical caustic. Sources of
contamination which limit observational SNR are considered in detail, including
the sun, the solar corona, the host star, and potential background objects. A
noise mitigation strategy of spectrally and spatially separating the light
using integral field spectroscopy is emphasized. A pseudoinverse-based image
reconstruction scheme demonstrates that direct reconstruction of an Earth-like
source from \textit{single} measurements of the Einstein ring is possible when
the critical caustic and observed SNR are sufficiently large. In this
arrangement, a mission would not require multiple telescopes or navigational
symmetry breaking, enabling continuous monitoring of the atmospheric
composition and dynamics on other planets.
