It is well-known that the spectra of the Gaudin model may be described in
terms of solutions of the Bethe Ansatz equations. A conceptual explanation for
the appearance of the Bethe Ansatz equations is provided by appropriate
$G$-opers: $G$-connections on the projective line with extra structure. In
fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced
version of opers called Miura opers; here, the opers appearing have only
regular singularities. Feigin, Frenkel, Rybnikov, and Toledano Laredo have
introduced an inhomogeneous version of the Gaudin model; this model
incorporates an additional twist factor, which is an element of the Lie algebra
of $G$. They exhibited the Bethe Ansatz equations for this model and gave a
geometric interpretation of the spectra in terms of opers with an irregular
singularity. In this paper, we consider a new approach to the study of the
spectra of the inhomogeneous Gaudin model in terms of a further enhancement of
opers called twisted Miura-Pl\”ucker opers and a certain system of nonlinear
differential equations called the $qq$-system. We show that there is a close
relationship between solutions of the inhomogeneous Bethe Ansatz equations and
polynomial solutions of the $qq$-system and use this fact to construct a
bijection between the set of solutions of the inhomogeneous Bethe Ansatz
equations and the set of nondegenerate twisted Miura-Pl\”ucker opers. We
further prove that as long as certain combinatorial conditions are satisfied,
nondegenerate twisted Miura-Pl\”ucker opers are in fact Miura opers.
