>>13441494Set
Then all cases are reduced via scaling to
.
So if you find solutions to this, you can invert the change of variables to find solutions of the family for in terms of
We have a local parameterization for positive solutions in all three numbers
and we can take all possible combinations of the above functions as other positive valued parameterizations. There are five other combinations that yield positive valued solutions. This is specifically for positive solutions. For solutions of one positive number and two negative numbers or two positive numbers and one negative number require a different parameterization.
