3D Picture of Schur Quartic and 64 lines

Question

The Schur Quartic, $x^4 -x y^3 = z^4 - z w^3$ has 64 lines.
The Clebsch Cubic, $w^3 + x^3 + y^3 + z^3 = (w+x+y+z)^3$ has 27 lines.

Work is required to show the Clebsch cubic nicely. Here's a more complicated form: $81 (x^3 + y^3 + z^3) - 189 (x^2 y + x^2 z + x y^2 + x z^2 + y^2 z + y z^2) + 54 x y z + 126 (x y + x z + y z) - 9 (x^2 + y^2 + z^2) - 9 (x + y + z) + 1 = 0$
With that, we can make a picture:

Can someone make a nice 3D picture of the Schur Quartic? Bonus if the 64 lines are shown.

Answer 1

As the comment says, a smooth real quartic has at most 56 real lines, and indeed there is a unique smooth real quartic with exactly 56 lines. These results are due to Degtyarev, Itenberg, and Sertöz:

Lines on quartic surfaces. Math. Ann. 368 (2017), no. 1-2, 753–809.

Here's a picture, of sorts, of that surface. Unfortunately only some of the lines are included:

Answer 2

Verifying the answer pointed at by Lazzaro. Showing 6 of the 56 lines. In Mathematica code, it's a ContourPlot of the following:

With[{w = 1}, 2 Sqrt[2] w (w - z) z (w + z) - x y (-3 w^2 + x^2 + y^2 - 3 z^2)] == 0 

.