Different forms of the double confluent Heun equation are studied. A generalized Riemann
scheme for these forms is given. An equivalent first-order system is introduced. This system can
be regarded from the viewpoint of the monodromy property. A corresponding Painlevé equation
is derived by means of the antiquantization procedure. It turns out to be a particular case of $P^3$.
A general expression for any Painlevé equation is predicted. A particular case of the Teukolsky
equation in the theory of black holes is examined. This case is related to the boundary between
spherical and thyroidal geometries of a black hole. Difficulties for its antiquantization are shown.