Is $C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2$ an embedded submanifold of $\mathbb{R}^2$? The problem
As a continuation of this question (where it was shown that $C$ was a closed $1$-dimensional submanifold for $c \neq 1/27$), I'm trying to find out whether or not $$C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2$$ is an embedded submanifold of $\mathbb{R}^2$ for $c = 1/27$.
What I have so far
I'm using the definitions from Spivak's A Comprehensive Introduction to Differential Geometry I:

-immersion: a differentiable function $f:M \rightarrow N$ s.t. $\text{rank } f = \dim M$, at all points of $M$,
-immersed submanifold: a subset $M_1$ of $M$ with a differentiable structure s.t. the inclusion map $i: M_1 \hookrightarrow M$ is an immersion,
-embedding: an injective immersion $f$ that is a homeomorphism onto its image,
-submanifold: an immersed submanifold $M_1 \subset M$ s.t. the inclusion map $i: M_1 \hookrightarrow M$ is an embedding.

So for $C$ to be an embedded submanifold, I take it it has to be a submanifold and it has to be an embedding.
My question is just regarding how to proceed in order to prove or disprove the above problem:
Should I consider the inclusion map $i: C \hookrightarrow \mathbb{R}^2$ and try to figure out whether or not this map is an immersion and an embedding?
And if it is, then try to figure out whether or not $C$ is an immersed submanifold (which would mean that it is a submanifold)?
Any help or hints on how to proceed is appreciated!
 A: $\newcommand{\RR}{{\mathbb R}}$
If you prove that the inclusion $i:C \to \RR^2$ is
1) an immersion
2) an embedding
then $C$ is by 1) an immersed submanifold and therefore by 2) a submanifold.
(The map $f$ here is always $i$).
In fact things are special here: There is a factorization:
$$x^3+x y + y^3 - 1/27 =  1/27\, \left( 3\,x-1+3\,y \right)  \left( 9\,{x}^{2}+3\,x-9\,xy+3\,y+1
+9\,{y}^{2} \right)$$
If you call the rightmost factor $g$ you can put it by an affine change of coordinates $x,y$ into the form $x^2+y^2$, so $g=0$ has a single real point, namely $x=-1/3, y= -1/3$.
The affine transformation is given concretely by
$$
\begin{align} 
x & =2/9\,\sqrt {3} \, y_1 - 1/3 \\
y & =1/3\,x_1+1/9\,\sqrt {3} \, y_1-1/3 
\end{align}
$$
That means, that as a set $C=\{(x,y)\mid 3 x + 3 y - 1 = 0\} \cup \{(-1/3,-1/3)\}$, a line in $\RR^2$ plus a point. So $C$ consists of two distinct components each of which is a submanifold, one of dimension $1$ and the other of dimension $0$. Adhering strictly to your definitions both together can not be called a submanifold, indeed $i: C \to \RR^2$ even fails to be an immersion.
