  Rigorous justification of the Hubbard-Stratonovich transformation for the
Pruisken-Sch\"afer type of parameterisations of real hyperbolic
O(m,n)-invariant domains remains a challenging problem. We show that a naive
choice of the volume element invalidates the transformation, and put forward a
conjecture about the correct form which ensures the desired structure. The
conjecture is supported by complete analytic solution of the problem for groups
O(1,1) and O(2,1), and by a method combining analytical calculations with a
simple numerical evaluation of a two-dimensional integral in the case of the
group O(2,2).
