Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
1. **Given Conditions**:
   - \(2xy\) is a perfect square.
   - \(x^2 + y^2\) is a prime number.

2. **Initial Observations**:
   - Let \(p = x^2 + y^2\). Since \(p\) is a prime number, \(p\) cannot be 2 because \(x^2 + y^2 \geq 2\) for any integers \(x\) and \(y\). Therefore, \(p > 3\).

3. **Parity Considerations**:
   - Since \(2xy\) is a perfect square, \(xy\) must be even. This implies that at least one of \(x\) or \(y\) must be even.

4. **Coprime Condition**:
   - Assume without loss of generality that \(x\) is even. Let \(d = \gcd(x, y)\). Since \(d \mid p\) and \(d < p\), it follows that \(d = 1\). Therefore, \(x\) and \(y\) are coprime.

5. **Representation of \(x\) and \(y\)**:
   - Since \(x\) is even and \(x\) and \(y\) are coprime, we can write \(x = 2a^2\) and \(y = b^2\) for some \(a, b \in \mathbb{N}\).

6. **Expression for \(p\)**:
   - Substituting \(x = 2a^2\) and \(y = b^2\) into \(p = x^2 + y^2\), we get:
     \[
     p = (2a^2)^2 + (b^2)^2 = 4a^4 + b^4.
     \]

7. **Factorization**:
   - We can factorize \(4a^4 + b^4\) as:
     \[
     4a^4 + b^4 = (2a^2 - 2ab + b^2)(2a^2 + 2ab + b^2).
     \]
   - Since \(p\) is a prime number, one of the factors must be 1. Therefore, we have:
     \[
     2a^2 - 2ab + b^2 = 1 \quad \text{or} \quad 2a^2 + 2ab + b^2 = 1.
     \]

8. **Solving the Equation**:
   - Consider \(2a^2 - 2ab + b^2 = 1\):
     \[
     2a^2 - 2ab + b^2 = 1 \implies a^2 + (a - b)^2 = 1.
     \]
   - Since \(a \geq 1\), the only solution is \(a = b = 1\).

9. **Conclusion**:
   - Substituting \(a = 1\) and \(b = 1\) back, we get \(x = 2a^2 = 2\) and \(y = b^2 = 1\). Therefore, \(p = 2^2 + 1^2 = 5\).

10. **Verification**:
    - The pairs \((x, y)\) are \((2, 1)\) and \((1, 2)\) since \(x\) and \(y\) are interchangeable in the problem.

The final answer is \( \boxed{ (x, y) = (2, 1) } \) and \((x, y) = (1, 2)\).