Contour lines and implicit function theorem

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$f = y^2-x^2+x^3$ I want to find all points $(x_0,y_0)$ such for some neighbor set $f(x,y) = f(x_0,y_0)$ is set $x=\varphi(y), \varphi \in C^{\infty}, \varphi(y_0)=x_0$.