  On a 3-manifold bounding a compact 4-manifold, let a conformal structure be
induced from a complete Einstein metric which conformally compactifies to a
K\"ahler metric. Formulas are derived for the eta invariant of this conformal
structure under additional assumptions. One such assumption is that the
K\"ahler metric admits a special K\"ahler-Ricci potential in the sense defined
by Derdzinski and Maschler. Another is that the K\"ahler metric is part of an
ambitoric structure, in the sense defined by Apostolov, Calderbank and
Gauduchon, as well as a toric one. The formulas are derived using the
Duistermaat-Heckman theorem. This result is closely related to earlier work of
Hitchin on the Einstein selfdual case.
