X$_1$=a$_1$+e$_1$,X$_2$=a$_2$+e$_2$, and X$_3$=a$_3$+e$_3$ where a$_1$+a$_2$+a$_3$=0.
e$_1$ has mean 0 and variance σ$_1$$^2$ e$_2$ has mean 0 and variance σ$_2$$^2$ e$_3$ has mean 0 and variance σ$_3$$^2$
X$_1$+X$_2$+X$_3$=a$_1$+a$_2$+a$_3$+e$_1$+e$_2$+e$_3$=0+e$_1$+e$_2$+e$_3$ is Gaussian
E(X$_1$+X$_2$+X$_3$)=0 Var(X$_1$+X$_2$+X$_3$)=σ$_1$$^2$+σ$_2$$^2$+σ$_3$$^2$
If you are given that X$_1$+X$_2$+X$_3$ =a ≠ 0, Set Z=X$_1$+X$_2$+X$_3$-a. Then Z=0.
E(a)=0 and Var(a)=σ$_1$$^2$+σ$_2$$^2$+σ$_3$$^2$ and a is Gaussian. So the question is how to split a into three parts s$_1$, s$_2$, s$_3$ such that s$_1$+s$_2$+s$_3$=a where s$_i$ is the amount X$_i$ is adjusted. Assume you want to
minimize E[(X$_1$-s$_1$)$^2$+(X$_2$-s$_2$)$^2$+(X$_3$-s$_3$)$^2$] where s$_1$+s$_2$+s$_3$=a. The question is how to choose s$_1$, s$_2$ and s$_3$ given a.
E[(X$_1$-s$_1$)$^2$+(X$_2$-s$_2$)$^2$+(X$_3$-s$_3$)$^2$] =
EX$_1$$^2$+E[s$_1$$^2$] + EX$_2$$^2$+E[s$_2$$^2$]+EX$_3$$^2$+E[s$_3$$^2$]=
E[s$_1$$^2$] +E[s$_2$$^2$]+E[s$_3$$^2$]=s$_1$$^2$+s$_1$$^2$+s$_3$$^2$.
Since s$_1$=a-s$_2$-s$_3$, s$_1$$^2$+s$_1$$^2$+s$_3$$^2$=(a-s$_2$-s$_3$)$^2$+s$_2$$^2$+s$_3$$^2$