Let $v,x,g$ be three vectors and $\alpha$ be a constant. The problem is
$$\min\limits_v \{\alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2\}$$
where $\|v\|^2=\sum\limits_{i=1}^{|v|}v_i^2$ and $|v|$ is the cardinality of $v$.
Its gradient is
$$\alpha v-x^Tvx+g^Tvg.$$
How should I get all vector $v^*$ such that $\alpha v^*-x^Tv^*x+g^Tv^*g=0$?
Obviously, there is no closed form solution. But what you can do is to solve the system $v=1/\alpha*(x^Tvx-g^Tvx)$. One way to solve is to iterative update $v$ from a initial point $v^0$. It is guarantee to converge but the result is only one local solution. If you want to obtain all $v$, it seems impossible.