Why are the solutions of the $3\times 3$ system like that? Consider the problem:$$
\min \quad -x_1^2-4x_1x_2-x_2^2\\ \text{s.t.} \quad x_1^2 + x_2^2 = 1$$
The KKT system is given by\begin{align*}
x_1 (-1 + v) + 2 x_2 &= 0 \tag{1} ,\\ x_2 (-1 + v) + 2 x_1 &= 0 \tag{2},\\ x_1^2 + x_2^2 &= 1 \tag{3}
\end{align*}
The solutions according to the book are$$
(x_1,x_2)=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right),\ v=3;\\
(x_1,x_2)=\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right),\ v=3;\\
(x_1,x_2)=\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right),\ v=-1,\\
(x_1,x_2)=\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right),\ v=-1,\\
$$
I tried to solved by hand: Substracting (1) and (2) I get $(x_1-x_2)(1+v)=0$. As $v$ cannot be zero (by the KKT condition) we have that $x_1=x_2$ if $v\neq -1$.
Thus I get the solutions:$$
(x_1,x_2)=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right);\\
(x_1,x_2)=\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right);\\
(x_1,x_2)=\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right);\\
(x_1,x_2)=\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right);\\
$$
only asking $v\neq -1$.
Why the book ask $v=-1$ and $v=3$? Could someone explain please?

First of all, your KKT system has a wrong sign, it must be
\begin{align*}
x_1 (-1 + v) \color{red}{-} 2 x_2 &= 0 \tag{1'} ,\\ x_2 (-1 + v) \color{red}{-} 2 x_1 &= 0. \tag{2'}
\end{align*}
However, the conclusion $(x_1-x_2)(1+v)=0$ from (1') minus (2') is correct.
You cannot get all four solutions "only asking $v\ne -1$" simply because if $v\ne-1$ then $x_1=x_2$, and it is only first two solutions that satisfy $x_1=x_2$. It means that when you study the case $v\ne-1$ you get first two solutions and $v=3$. 
For a complete solution, you have to study the remaining case $v=-1$ as well. Set $v=-1$ into (1') and (2') to get $x_1+x_2=0$, i.e. $x_1=-x_2$, which gives you another two solutions from (3).
