Realizing a set as the image of a smooth map

Question

Consider the following subset of $\mathbb{R}^2$:

$S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$

(See here for a plot on Wolfram Alpha.)

The origin $(0, 0)$ is a kind of singular point of $S$.

Question: Does there exist a $C^2$ map $\varphi \colon M \to \mathbb{R}^2$ on some manifold $M$ such that $\varphi(M) = S$?

Further context:

• I would like for $M$ to be finite dimensional.
• Ideas of techniques to construct or disprove the existence of such $\varphi$ are welcome.
• It is satisfactory if the image of $\varphi$ is locally the same as $S$ around the origin.
• The set $S' = \{ (x, y) : y^2 \leq x^3 \}$ is the right-half of $S$; it admits a smooth lift: $\varphi(s, t) = (s^2, s^3 \sin(t))$.

Answer 1

$(s^4\, \text{sign}(s),s^6 \sin t)$.

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Or even an infinitely smooth map:

$$(s^2 e^{-2/|s|}\, \text{sign}(s),s^3 e^{-3/|s|} \sin t),$$ with $(0,0)$ for $s=0$.