= Hamiltonian perturbation preserving one regular level
{c}
{title2=$K=H+\tfrac12\sum_jG_j^2$}
For independent integrals $G_1=H,G_2,\ldots,G_r$, the <Hamiltonian> $K=H+\frac12\sum_jG_j^2$ agrees with $H$ to first order on the common zero level. On a nonempty level $G=c\ne0$, the vector-field difference is $\sum_jc_jX_{G_j}$ and is nowhere zero by independence of the <differentials>. Nonemptiness is necessary if one expresses the conclusion as inequality between restricted <vector fields>.
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