Hamiltonian perturbation preserving one regular level ($K=H+\frac{1}{2}{\sum }_j{G}_j^2$)

For independent integrals $G_1=H,G_2,\dots ,G_r$, the Hamiltonian $K=H+\frac{1}{2}{\sum }_j{G}_j^2$ agrees with $H$ to first order on the common zero level. On a nonempty level $G=c\ne 0$, the vector-field difference is ${\sum }_j{c}_j{X}_{G_j}$ and is nowhere zero by independence of the differentials. Nonemptiness is necessary if one expresses the conclusion as inequality between restricted vector fields.