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$$\varphi(u,v)=\left( \begin{array}{c} \dfrac{u}{3}\left(1-\dfrac{u^2}{3}+v^2\right)\\ -\dfrac{v}{3}\left(1-\dfrac{v^2}{3}+u^2\right)\\ u^2-v^2 \end{array} \right)\qquad \begin{array}{l} u\in\mathbb{R}\\ v\in\mathbb{R} \end{array} $$