  By using pairs of nontrivial rational solutions of congruent number equation
$$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right
(Pythagorean) triangles with one common side and the other sides equal to the
sum and difference of the squares of the same rational numbers. The
parametrizations are found for following Diophantine systems: \begin{align*}
(p^2\pm q^2)^2-a^2 & =\square_{1,2}\,, \\[0.2cm] c^2-(p^2\pm q^2)^2 &
=\square_{1,2}\,, \\[0.2cm] a^2+(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm]
(p^2\pm q^2)^2-a^2 & =(r^2\pm s^2)^2. \end{align*}
