The issue I discussed in this thread. Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$
Generally speaking at the forum often ask a question about this equation. So I think that will not solve different each time Diophantine equation is better to write the equation in this General form:
$a,b,c,e,j,t - $ integer coefficients which are defined by the problem statement.
The task is simple - to write a formula describing the parameterization of the equation. The formula itself and will specify conditions when possible integer solutions.
Many people like Diofantos geometry, but its methods are known for a very long time - here is inefficient. It is always better to have a single formula describing all equations than every time to solve the new equation.