Integer solutions for $a+b+c+d=0$, $a^3+b^3+c^3+d^3=24$ How to get all integer solutions for $$a+b+c+d=0,$$ and $$a^3+b^3+c^3+d^3=24$$
So far I've put $a=-b-c-d$ into 2nd equation and try to factorise it, but didn't find anything useful.

Since $d=-a-b-c$,  you can  check the following identity:
$$a^3+b^3+c^3+d^3=-3(a+b)(a+c)(b+c).$$ 
Thus $$(a+b)(a+c)(b+c)=-8,$$
and that easily gives you all integer solutions. For instance, the case
\begin{cases}
a+b=1 \\
a+c=1\\
b+c=-8 
\end{cases}
gives you $(a,b,c,d)=(5,-4,-4,3),$  and  the case
\begin{cases}
a+b=-2 \\
a+c=-2\\
b+c=-2 
\end{cases}
gives you $(a,b,c,d)=(-1,-1,-1,3).$
