Find all $c\in \mathbb{R}$ such that $f^{-1}(c)$ is a submanifold(Solution verification) Consider the following problem from my course notes on manifolds:

Question:Define $f: \mathbb{R}^2 \to \mathbb{R}$  by $f(x,y)= x^3 -6xy+y^2$. Find all values $c\in \mathbb{R}$ such that $f^{-1}(c)\in M$ is an embedded submanifold of $\mathbb{R}^2$.

So, $f$ is an onto map (just take $y=0$ , $x^3 $ is onto) and $f^{-1}(c) = ${$(x,y): f(x,y)=c$}. $\mathbb{R}^3$ is a manifold and there always exists a chart $(U,\phi)$ , $\phi$ is homeo. So, I think that using onto property of $f$ I think for all $c\in \mathbb{R}$ , $f^{-1}(c)$ is a submanifold as $\phi( U \cap f^{-1} (c)) $ will always be in  $\mathbb{R}^3$ and hence satisfying definition of submanifold.
Am I right?

FedericoFallucca's answer shows that for $c \in \Bbb R \setminus \{0,-108\}$, $f^{-1}(c)$ is a submanifold of $\Bbb R^2$ with codimension $1$. But it does not prove that $f^{-1}(0)$ neither $f^{-1}(108)$ is or isn't a submanifold: it just shows that for these values, the Regular value Theorem does not apply.
The consideration about the critical values $0$ and $-108$ is more intricate.
Idea to show that $f^{-1}(0)$ isn't a submanifold.
Note that the two paths
\begin{align}
\gamma_1(t) &= (t, 3t + \sqrt{9t^2-t^3})\\
\gamma_2(t) &= ( t, 3t - \sqrt{9t^2-t^3})
\end{align}
defined on a suitable open interval $(-\varepsilon,0]$ are of class $\mathcal{C}^1$. With a bit of calculus, we find that $\gamma_1'(0) = (1,0)$ while $\gamma_2'(0) = (1,6)$. Therefore, if $f^{-1}(0)$ were a submanifold, its tangent space at $(0,0)$ would be two dimensional: so would be the dimension of $f^{-1}(0)$ in a neighbourhood of the origin. Hence, $f^{-1}(0)$ should contain a neighbourhood of $(0,0)$, which is plain wrong.
Here is a discussion about $f^{-1}(-108)$.
It depends on the definition of submanifold you consider. Note that $f^{-1}(-108)$ has two connected components: one is a smooth curve contained in the half plane $x<0$, the other is the point $\{(6,18)\}$. Hence, each of the connected component are submanifolds with respective dimension $1$ and $0$. If you allow submanifolds to have several components with different dimensions, then it is indeed a submanifold.
In conclusion:

*

*If one allow submanifolds to have several connected components with different dimensions, then the answer is $c \in \Bbb R \setminus \{0\}$.

*If not, the answer is $\Bbb R \setminus \{0,-108\}$.

Comment: why does $f$ behave that differently regarding the critical values $0$ and $-108$? The answer is that $(0,0)$ is a saddle point while $(6,18)$ is a local minimum, unique in a neighbourhood.
