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One can exchange $S$ with $I$ to obtain a new Lagrangian $\\tilde{\\cal L}$ and a new Noether current $\\tilde{\\bf J}=\\exp(2I)\\nabla S$, conserved under the equations of motion of $\\tilde{\\cal L}$. However this new current $\\tilde{\\bf J}$ is generally not conserved under the equations of motion of the original Lagrangian ${\\cal L}$. 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