  In this paper we reconsider the original Kolmogorov normal form algorithm
with a variation on the handling of the frequencies. At difference with respect
to the Kolmogorov approach, we do not keep the frequencies fixed along the
normalization procedure. Besides, we select the frequencies of the final
invariant torus and determine a posteriori the corresponding starting ones. In
particular, we replace the classical translation step with a change of the
frequencies. The algorithm is based on the original scheme of Kolmogorov, thus
exploiting the fast convergence of the Newton-Kantorovich method.
