  We present a non-standard eigenvalue problem that arises in the linear
stability of a three-layer Hele-Shaw model of enhanced oil recovery. A
nonlinear transformation is introduced which allows reformulation of the
non-standard eigenvalue problem as a boundary value problem for Kummer's
equation when the viscous profile of the middle layer is linear. Using the
existing body of works on Kummer's equation, we construct an exact solution of
the eigenvalue problem and provide the dispersion relation implicitly through
the existence criterion for the non-trivial solution. We also discuss the
convergence of the series solution. It is shown that this solution reduces to
the physically relevant solutions in two asymptotic limits: (i) when the linear
viscous profile approaches a constant viscous profile; or (ii) when the length
of the middle layer approaches zero.
