For D somewhere between -11 and -12 (I think!), the following function produces a graph with a closed curve and two singular points.
$$ y^4-4y^2=-x^4+3x^3+3x^2+Dx-3 $$
I plotted a 2D graph, animated for various values of D. The two singular point can be thought of as local peaks in the 3D space (x, y, D).
What is the exact value of D when this occurs?
On
Using Mathematica, I obtained the approximate value $$D \approx -11.960949537899285187548631685245893662483813189418\ldots.$$ This is a root of the quartic polynomial $$27 z^4+270 z^3-3591 z^2+2898 z+457807.$$ This value corresponds to a curve that self-intersects at $y = \pm \sqrt{2}$ and $x \approx 1.081941108085261568294448553915$. But if you want the value of $D$ for which there are two isolated points, what you want is a different root of the same quartic above: $$D \approx -13.658381026615799846061572151507704638149295780586\ldots.$$
However, there is a difference in the equation you stated, and the equation given in the video link. Yours says $+3$ whereas the video says $-3$. If we go by the video, then the desired values of $D$ correspond to the real roots of the quartic $$27 z^4+270 z^3-675 z^2-3042 z+9751.$$ These are approximately $$\begin{align*} D &\approx -11.072841028502857454702305258829734721106264530500\ldots, \\ D &\approx -4.0881318682078769707577137157340189215903558111322\ldots, \end{align*}$$ the first of which is the situation with two isolated points. The coordinates of these points are approximately $(x,y) = (2.3934900075826234528, \pm \sqrt{2})$.
After correction of the typo, the singular points are shown on the joints graphs.
Figure 3 shows the case of isolated singular points.