Perform a rotation $(a,b,c)=\left(x\cos\frac{\pi}{4}-y\sin\frac{\pi}{4},x\sin\frac{\pi}{4}+y\cos\frac{\pi}{4},z\right)=\left(\frac{x-y}{\sqrt{2}},\frac{x+y}{\sqrt{2}},z\right)$. This gives the system $x^2+y^2+z^2=1$ and $|x|=1$. The new system then has the solutions $(\pm 1,0,0)$, so that the original system has solutions $\pm\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0\right)$.