I posted two solutions to About some Pythagorean Quadruples
The second answer covers the situation that the OP says is of interest, ruling out values of her variable $K$ for which there is no solution to her Diophantine equation.
The OP repeatedly emails me, ignoring or misunderstanding what I have already said.
I have asked the OP to stop contacting me, in my final email and in a comment at the question. I have forwarded the (long) email thread to Mariano and Willie, moderators I know through MO.
Any assistance will be appreciated. I am not sure what in the emails can be trusted, with the exception that i think it likely the OP is Bulgarian.