Integer points on an ellipsoid

Take the equation mod 7:

$\begin{aligned} x^2-2x &\equiv 6 \pmod{7} \\ x &\equiv 1 \pmod{7} \end{aligned}$

Take the equation mod 5:

$\begin{aligned} y^2-2y &\equiv 4 \pmod{5} \\ y &\equiv 1 \pmod{5} \end{aligned}$

Take the equation mod 3:

$\begin{aligned} z^2-2z &\equiv 2 \pmod{3} \\ z &\equiv 1 \pmod{3} \end{aligned}$

If there exist integer points on the ellipsoid, the coordinates must satisfy these congruences. Now convert the equation to standard form:

$\frac{(x-4.5)^2}{63}+\frac{(y-3.5)^2}{45}+\frac{(z-1)^2}{13.5}=1$

This gives the minimum and maximum values of the variables:

$\begin{aligned} x &\in [4.5-\sqrt{63},4.5+\sqrt{63}] \\ y &\in [3.5-\sqrt{45},3.5+\sqrt{45}] \\ z &\in [1-\sqrt{13.5},1+\sqrt{13.5}] \\ \end{aligned}$

The maximum integer coordinates that satisfy the congruences and intervals above are $(8,6,4).$ Plugging this in confirms that this point is on the ellipsoid. Therefore, the maximum value of $a+b+c$ is $8+6+4=\boxed{18}.$