  We consider Keldysh-type operators, $ P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q
(x, D_{x'} ) $, $ x = ( x_1, x') $ with analytic coefficients, and with $ Q (
x, D_{x'} ) $ second order, principally real and elliptic in $ D_{x'} $ for $ x
$ near zero. We show that if $ P u =f $, $ u \in C^\infty $, and $ f $ is
analytic in a neighbourhood of $ 0 $ then $ u $ is analytic in a neighbourhood
of $ 0 $. This is a consequence of a microlocal result valid for operators of
any order with Lagrangian radial sets. Our result proves a generalized version
of a conjecture made by the second author and Lebeau and has applications to
scattering theory.
