I have a polynomial system with three equations and three unknowns i wish to solve analytically. I can obtain a numerical solution easily but for my project i need to find a analytical solution. The system is $$(\frac{\sqrt3}{2}x-\frac{1}{2}y+a)^2+z^2=r_{1}^2$$ $$(-\frac{\sqrt3}{2}x-\frac{1}{2}y+a)^2+z^2=r_{2}^2$$ $$(y+a)^2+z^2=r_{3}^2$$
where $a$ and the $r_{i}$ are constants.
I think the problem can be stated as the intersection between three elliptical cylinders. But I don't know if that helps or how to proceed from there.
Let us change variables defining $$X=\frac{\sqrt 3}2 x \qquad\qquad Y=\frac 12 y -a\qquad \qquad Z=z^2$$ to make $$(X-Y)^2+Z=r_1^2\tag 1$$ $$(X+Y)^2+Z=r_2^2\tag 2$$ $$(2Y+a)^2+Z=r_3^2\tag 3$$
Subtracting $(1)$ for $(2)$ gives $$X=\frac{r_2^2-r_1^2}{4 Y}\tag 4$$ and $(3)$ gives $$Z=r_3^2-(2Y+a)^2\tag 5$$ All of that makes $$48 Y^4+64 a Y^3+8 \left(2 a^2+r_1^2+r_2^2-2 r_3^2\right)Y^2-(r_1^4+r_2^4-2 r_1^2 r_2^2)=0\tag 6$$ that you can solve with radicals (have a look here and have fun !).
You will have at most four real roots but, more than likely, some of them will be discarded because of $(5)$ ($Z$ must be positive).
When solved, go back to $(x,y,z)$.