The complete solution to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$, for $k=1,2$? It's quite easy to give the complete rational solution to,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$
One can express it in the form,
$$(p+q)^k+(r+s)^k+(t+u)^k=(p-q)^k+(r-s)^k+(t-u)^k\tag{2}$$
and impose 2 linear conditions on $p,q,r,s,t,u$. However, an alternative is,
$$(ad+e)^k+ (bc+e)^k+ (ac+bd+e)^k = (ac+e)^k + (bd+e)^k + (ad+bc+e)^k\tag{3}$$
which is already identically true for $k=1,2$. 
Question: How do you prove $(3)$ is the complete rational solution to $(1)$? 
(I actually have a proof, but was wondering if someone else has a more elegant way to do it.) 
 A: With different notation for the main variables, the system is $u+v+w=x+y+z$ and the squared equation $u^2+v^2+w^2=x^2+y^2+z^2.$ Your substitutions are
$$ad+e=u,\\ bc+e=v,\\ac+bd+e=w,\\ ac+e=x,\\ bd+e=y,\\ ad+bc+e=z.$$
From this two expressions for $e$ are $u+v-z$ and $x+y-w.$ These are compatible because of the linear equation of the system. Now one can solve for the pairwise products of your substitution (using $e=u+v-z=x+y-w$) as
$$ad=z-v,\\ bc=z-u,\\ ac=w-y,\\ bd=w-x.\tag{1}$$
This implies that for your substitution to cover all solutions, it must be the case, since $(ad)(bc)=(ac)(bd),$ that
$$(z-u)(z-v)=(w-x)(w-y).\tag{2}$$
I haven't been able to show this is a consequence of the system consisting of the linear and quadratic equations in $u,v,w,x,y,z$. However it seems to hold for the one example I looked at, namely $(1,5,6,2,3,7).$ No matter how I set up the variables the relation (2) was OK.
Assuming that (2) holds, there is the question of how to recover $a,b,c,d$, since we already know $e$ and the remaining parameters only appear in products. There are relations like $[(ad)(ac)]/[(bc)(bd)]=(a/b)^2$ which imply that certain ratios of the differences of the variables are squares; even this seemed to hold in the simple example I looked at. It seems these values of $a,b,c,d$ are not unique, but can be found by choosing say $a$ at random, and then using the product expressions from (1) to obtain $b,c,d$. 
In conclusion, if the relation (2) can somehow be shown to follow from the initial system (even if one must suppose a certain ordering of the main variables), it seems your parametrization should cover all solutions.
A: Complementing the proof by ckrao in the link given by Myerson, here is mine. The general formula for,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$
where $x_1 \ne y_1$ can be given as,
$$(p+q)^k+(r+s)^k+(t+u)^k=(p-q)^k+(r-s)^k+(t-u)^k\tag{2}$$
with two linear conditions. Expanding at $k=1,2$, it must be the case that,
$$q+s+u =0$$
$$pq+rs+tu=0$$
and then one can linearly solve for $t,u$.  Alternatively,
$$(ad+e)^k+ (bc+e)^k+ (ac+bd+e)^k = (ac+e)^k + (bd+e)^k + (ad+bc+e)^k\tag{3}$$
Proof:
Equating terms of $(2)$ and $(3)$,
$$\begin{aligned}
p+q &= ad+e\\
r+s &= bc+e\\
p-q &= ac+e\\
r-s &= bd+e\\
\end{aligned}\tag{4}$$
we can also linearly solve for $a,b,c,e$ (the exact forms are tedious to write down here). But it can then be observed that $a,b,c,e$ and $t,u$ satisfy,
$$\begin{aligned}
t+u &= ac+bd+e\\
t-u &= ad+bc+e\\
\end{aligned}\tag{5}$$
thus proving $(2)$ (with the two linear conditions) and $(3)$ are equivalent. (End proof.)
P.S. I was hoping for a proof that could derive $(3)$ from first principles, not the sketch that I gave above since it needs prior knowledge of $(3)$.
