<s>[INST] Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions. 
Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

I am not able to find an idea on how to proceed with the above questions. I have found only the obvious solution $(1,1,1)$.
Could you please provide some hints and ideas on how to proceed with the above question? Also, can we find the solutions?
Thanks.
 [/INST] Let $y=1+a, z=1-a$. Then 
$$x^3+2(1+3a^2)=2(1+a^2)+x^2$$
$$x^2-x^3=4a^2.$$
Let $1-x=4p^2$, then 
$$x^2(1-x)=(4p^2-1)^24p^2=(2a)^2.$$
Let $a=p(4p^2-1)$. Then
$$(x,y,z)=(1-4p^2, 1+p(4p^2-1), 1-p(4p^2-1))$$
</s>