When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
I have observed that the anomaly disappears when the resolution is explicitly asked over the integers (the qualifier Diophantine also works).
So it seems that the flaw would be in the logics for the "Integer solutions" section that comes along with an unspecified domain (presumably $\mathbb C$ by default). As suggested by @chiphurst, this could be because a general solver is used and might fail to find the exact integers.