= Solution
Let
$$
F(x,y)=x^3y+y^3+x.
$$
Its complex partial derivatives are
$$
F_x=3x^2y+1,
\qquad
F_y=x^3+3y^2.
$$
There is no point of $X'$ at which both vanish. Indeed, if $F=F_x=F_y=0$, then $F_y=0$ and $F=0$ give
$$
x^3=-3y^2,
\qquad
x=2y^3.
$$
Neither $x$ nor $y$ can be zero because $F_x=0$. Substitution into $F_x=0$ gives $12y^7+1=0$, whereas substitution into $F_y=0$ gives $8y^7+3=0$; these equations are incompatible.
For every $p=(x_0,y_0)\in X'$, at least one partial derivative is nonzero. If $F_y(p)\ne0$, the <holomorphic implicit function theorem> expresses the curve near $p$ as $y=g(x)$, and the restriction of the $x$-projection is a holomorphic coordinate chart. If $F_x(p)\ne0$, use the $y$-projection instead. On overlaps, the transition functions are restrictions of holomorphic coordinate functions and are holomorphic. These charts define an atlas making $X'$ a one-dimensional <complex manifold>, hence a <Riemann surface>.
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