It is well-known that every positive integer can be written as the sum of the squares of four integers. More nuanced patterns are also known — for example, every odd positive integer has a representation as the sum of the squares of four integers which sum to $1$.
Evidently, every positive integer can also be written as the sum of the squares of eight integers (the trivial case being the addition of four $0^2$ terms, but non-trivial representations usually exist as well).
This leads me to the following question:
QUESTION: Are there any known “interesting” patterns for the representations of positive integers as the sum of the squares of eight integers?