  We show a general method allowing the solution calculation, in the form of a
power series, for a very large class of nonlinear Ordinary Differential
Equations (ODEs), namely the real analytic $\sigma\pi$-ODEs (and, more in
general, the real analytic $\sigma\pi$-{\it reducible} ODEs) in many
indeterminates, characterized by an ODE-function given by generalized
polynomials of the indeterminates and their derivatives, i.e. functions
formally polynomial with exponents, though the exponent can be any real number.
The solution method consists in reducing the ODE to a certain canonical
quadratic ODE, with a larger number of indeterminates, whose solutions include,
as sub-solutions, the original solutions, and for which a recursive formula is
shown, giving all coefficients of the solution power series directly from the
ODE parameters. The reduction method is named {\it exact quadratization} and
was formerly introduced in another our article, where we considered explicit
ODEs only. In the present paper, which is self-contained to a large extent, we
review and complete the theory of exact quadratization by solving issues, such
as for instance the {\it globality} of the quadratization, that had remained
open, and also extend it to the more general case of an implicitly defined ODE.
Finally, we argue that the result can be seen as a partial solution of a
differential version of the 22nd Hilbert's problem.
