Two diophantine equations with lots of unknowns

Question

Is it possible (tractable) to determine if the following system of equations has any nontrivial solutions (ie, none of the unknowns are zero) in the domain of integers?

$$A^2 + B^2=C^2 D^2$$ $$2 C^4 + 2 D^4 = E^2 + F^2$$

Answer 1

for the second one, take $C > D > 0,$ then $$ E = C^2 - D^2, \; \; \; F = C^2 + D^2 $$

If you wanted a system, take any $C,D \equiv 1 \pmod 4$ distinct primes, such as $5,13.$ We get the Pythagorean triple $16^2 + 63^2 = 65^2 = 5^2 13^2.$ Then $2 \cdot 5^4 + 2 \cdot 13^4 = (13^2 - 5^2)^2 + (13^2 + 5^2)^2 = 144^2 + 194^2.$

Answer 2

To solve,

$$A^2+B^2=C^2 D^2\\ (2C)^4+(2D)^4=E^2+F^2$$

Choose,

$$\begin{aligned} A&=2(ac-bd)(ad+bc)\\ B&=(ac-bd)^2-(ad+bc)^2\\ C&=a^2+b^2\\ D&=c^2+d^2\\ E&=(a^2+b^2 )^2-(c^2+d^2 )^2\\ F&=(a^2+b^2 )^2+(c^2+d^2 )^2\\ \end{aligned}$$