  We study the {\it regular} action of an analytic pseudo-group of
transformations on the space of germs of various analytic objects of local
analysis and local differential geometry. We fix a homogeneous object $F_0$ and
we are interested in an analytic normal form for the whole affine space $\{F_0
+ h.o.t.\}$. We prove that if the cohomological operator defined by $F_0$ has
the big denominators property and if a formal normal form is well chosen then
this formal normal form holds in analytic category. We also define big
denominators in systems of nonlinear PDEs and prove a theorem on local analytic
solvability of systems of nonlinear PDEs with big denominators. %This theorem,
which a priori has nothing to do with local classification problems,generalizes
the theorem on analytic normal forms. Moreover, we prove that if the
denominators grow "relatively fast", but not fast enough to satisfy the big
denominator property, then we have a normal form, respectively local
solvability of PDEs, in a formal Gevrey category. We illustrate our theorems by
explanation of known results and by new results in the problems of local
classification of singularities of vector fields, non-isolated singularities of
functions, tuples of germs of vector fields, local Riemannian metrics and
conformal structures.
