Is the following True or False?

The simplest (and most boring) solution is just $(0,0,0,0)$ . A simple infinite family of solutions is $(3x^2,3x^2,3x^2,9x^3)$ .

If we want the integers to be distinct, then $(1,2,3,6)$ is the simplest solution. If $(a,b,c,d)$ is a solution, then so is $(ax^2,bx^2,cx^2,dx^3)$ , so there are infinitely many solutions with $(a,b,c,d)$ distinct.

If we're feeling really picky, we might want to insist that $\gcd{(a,b,c,d)}=1$ . There seem to be lots of solutions satisfying this additional criterion, but I haven't yet found a rule to generate them - any suggestions?

Assuming we want non-zero integers, one example is $(a,b,c,d) = (1,2,3,6)$ . There are many others

Is numerical example enough here? I can choose $0^3+1^3+2^3= 3^2$ . Also, this solution is not unique.