nt.number theory - A property of primes congruent to $7pmod 8$ expressed as sums of four squares.

Too long for a comment. Your problem reminds me of this. If anything directly helpful comes to mind on your problem I will let you know of course.

Please allow me to draw your attention to the fascinating Markov-Hurwitz Diophantine equation
$$ x_1^2 + x_2^2 + cdots + x_n^2 = a ; x_1 x_2 ldots x_n $$
in positive integers, with
$$ 1 leq a leq n $$
as shown by Hurwitz (1907).

See my answer to
Numbers characterized by extremal properties
especially the Markov tree, Markov (1880)
http://en.wikipedia.org/wiki/Markov_number

If, for example, $x_1$ is fairly large , it can be replaced
by $ a x_2 x_3 ldots x_n - x_1 $ to give another solution with smaller values. This process can be repeated until one arrives at a "fundamental solution" which satisfies a certain inequality: ordered so that $x_1$ is indeed the largest, a fundamental solution has
$$ 2 x_1 leq a x_2 x_3 ldots x_n .$$

So a fundamental solution is the root of a tree of solutions for fixed pair $(n,a).$ The first time that a pair $(n,a)$ requires a disconnected forest is $(n=14, a=1)$ one tree with (decreasingly ordered) root
$(6,4,3,1,1,ldots)$ and another tree with ordered root $(3,3,2,2,1,1,ldots).$

So many things...my conjecture that, for a fundamental solution in nonincreasing order,
$ 5 x_1^2 leq 9 ( n+6) .$ Finally the right hand side $ a ; x_1 x_2 ldots x_n $ can be replaced by any of those symmetric polynomials where all exponents are at most one, as we still get trees.