  We consider a one-parameter family of degenerately elliptic operators
$\cal{L}_\gamma$ on the closed disk $\mathbb{D}$, of Keldysh (or Kimura) type,
which appears in prior work [Mishra et al., Inverse Problems (2022)] by the
authors and Mishra, related to the geodesic X-ray transform. Depending on the
value of a constant $\gamma\in \mathbb{R}$ in the sub-principal term, we prove
that either the minimal operator is self-adjoint (case $|\gamma|\ge 1$), or
that one may construct appropriate trace maps and Sobolev scales (on
$\mathbb{D}$ and $\mathbb{S}^1=\partial\mathbb{D}$) on which to formulate
mapping properties, Dirichlet-to-Neumann maps, and extend Green's identities
(case $|\gamma|<1$). The latter can be reinterpreted in terms of a boundary
triple for the maximal operator, or a generalized boundary triple for a
distinguished restriction of it. The latter concepts, object of interest in
their own right, provide avenues to describe sufficient conditions for
self-adjointness of extensions of $\cal{L}_{\gamma,min}$ that are parameterized
in terms of boundary relations, and we formulate some corollaries to that
effect.
