Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
1. Given the system of equations:
   \[
   5x \left( 1 + \frac{1}{x^2 + y^2}\right) = 12
   \]
   \[
   5y \left( 1 - \frac{1}{x^2 + y^2} \right) = 4
   \]

2. We can rewrite these equations as:
   \[
   x + \frac{x}{x^2 + y^2} = \frac{12}{5} \quad \cdots (1)
   \]
   \[
   y - \frac{y}{x^2 + y^2} = \frac{4}{5} \quad \cdots (2)
   \]

3. Introducing the complex number \( z = x + iy \) where \( i = \sqrt{-1} \), we know that \( z \overline{z} = x^2 + y^2 \).

4. Adding equations (1) and (2) multiplied by \( i \):
   \[
   x + iy + \frac{x - iy}{x^2 + y^2} = \frac{12}{5} + i \cdots \frac{4}{5}
   \]
   \[
   z + \frac{1}{z} = \frac{12}{5} + i \cdots \frac{4}{5}
   \]

5. Let \( t = \frac{2}{5}(3 + i) \). Then:
   \[
   z + \frac{1}{z} = 2t
   \]

6. Solving the quadratic equation for \( z \):
   \[
   z^2 - 2tz + 1 = 0
   \]
   \[
   z = t \pm \sqrt{t^2 - 1}
   \]

7. Calculating \( t^2 - 1 \):
   \[
   t = \frac{2}{5}(3 + i) = \frac{6 + 2i}{5}
   \]
   \[
   t^2 = \left( \frac{6 + 2i}{5} \right)^2 = \frac{36 + 24i - 4}{25} = \frac{32 + 24i}{25}
   \]
   \[
   t^2 - 1 = \frac{32 + 24i}{25} - 1 = \frac{32 + 24i - 25}{25} = \frac{7 + 24i}{25}
   \]

8. Taking the square root:
   \[
   \sqrt{t^2 - 1} = \sqrt{\frac{7 + 24i}{25}} = \frac{4 + 3i}{5}
   \]

9. Therefore:
   \[
   z = \frac{6 + 2i}{5} \pm \frac{4 + 3i}{5}
   \]
   \[
   z = \frac{10 + 5i}{5} = 2 + i \quad \text{or} \quad z = \frac{2 - i}{5}
   \]

10. Comparing the real and imaginary parts:
    \[
    x = 2, y = 1 \quad \text{or} \quad x = \frac{2}{5}, y = -\frac{1}{5}
    \]

11. Verifying the solutions:
    - For \( (x, y) = (2, 1) \):
      \[
      5 \cdot 2 \left( 1 + \frac{1}{2^2 + 1^2} \right) = 10 \left( 1 + \frac{1}{5} \right) = 10 \cdot \frac{6}{5} = 12
      \]
      \[
      5 \cdot 1 \left( 1 - \frac{1}{2^2 + 1^2} \right) = 5 \left( 1 - \frac{1}{5} \right) = 5 \cdot \frac{4}{5} = 4
      \]

    - For \( (x, y) = \left( \frac{2}{5}, -\frac{1}{5} \right) \):
      \[
      5 \cdot \frac{2}{5} \left( 1 + \frac{1}{\left( \frac{2}{5} \right)^2 + \left( -\frac{1}{5} \right)^2} \right) = 2 \left( 1 + \frac{1}{\frac{4}{25} + \frac{1}{25}} \right) = 2 \left( 1 + \frac{1}{\frac{5}{25}} \right) = 2 \left( 1 + 5 \right) = 12
      \]
      \[
      5 \cdot \left( -\frac{1}{5} \right) \left( 1 - \frac{1}{\left( \frac{2}{5} \right)^2 + \left( -\frac{1}{5} \right)^2} \right) = -1 \left( 1 - \frac{1}{\frac{5}{25}} \right) = -1 \left( 1 - 5 \right) = 4
      \]

The final answer is \(\boxed{(x, y) = (2, 1), \left( \frac{2}{5}, -\frac{1}{5} \right)}\).