  In [Grenier-Nguyen], we introduced so called {\em generators} functions to
precisely follow the regularity of analytic solutions of Navier Stokes
equations. In this short note, we give a presentation of these generator
functions and use them to give existence results of analytic solutions to some
classical equations, namely to hyperbolic equations, to incompressible Euler
equations, and to hydrostatic Euler and Vlasov models. The use of these
generator functions appear to be an alternative way to the use of the classical
abstract Cauchy-Kovalevskaya theorem [Asano,Caflisch,Nirenberg,Safonov].
