Solving this $4$ variable diophantine equation Is there a way to solve this Diophantine equation in $a,b,c,d$? $$19a^3-33a^2b+3a^2c+30a^2d+21ab^2+24abc-12abd-15ac^2-54acd-30ad^2+ $$ $$2b^3-12b^2c-6b^2d+42bc^2+108bcd+60bd^2-7c^3-51c^2d-99cd^2-56d^3=0$$
Wolfram Alpha unfortunately cannot understand my input whenever I input this equation, and I know no strategy to solve these kinds of equations. So basically, I'm stuck...

I want just all possible values for $a,b,c,d$ just like what Wolfram Alpha does.
Note: $a\neq b\neq c\neq d\neq 0$

As the OP pointed out in the comments that the equation is equivalent to,
$$(a + 2c - 2b + 3d)^3 + (2a - b + 2c + 4d)^3 - (a + b + 2c + 2d)^3 - (3a - 2b+ c + 3d)^3 = 0$$
then by equating terms, its complete solution is,
$$\begin{aligned}
a= -2 x_1 - 7 x_2 - 5 x_3 - 8 x_4\\
b=-6 x_1 + 5 x_2 - 2 x_3 + 2 x_4\\
c=9 x_1 - 14 x_2 - 10 x_3 - 3 x_4\\
d=-5 x_1 + 15 x_2 + 7 x_3 + 6 x_4
\end{aligned}$$
where,
$$x_1^3+x_2^3+x_3^3+x_4^3 = 0$$
The complete rational solution by Euler is well-known.
