Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases}  x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2  \end{cases}$$
1. **Add the first and the last equations:**
   \[
   (x_1 + x_2) + (x_3 + x_4) = (x_3^2 + x_4^2 + 6x_3x_4) + (x_1^2 + x_2^2 + 6x_1x_2)
   \]
   Simplifying, we get:
   \[
   x_1 + x_2 + x_3 + x_4 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + 6(x_1x_2 + x_3x_4)
   \]

2. **Generalize for any permutation \(\sigma\):**
   For any permutation \(\sigma: \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}\), we have:
   \[
   x_{\sigma(1)} + x_{\sigma(2)} = x_{\sigma(3)}^2 + x_{\sigma(4)}^2 + 6x_{\sigma(3)}x_{\sigma(4)}
   \]
   This implies:
   \[
   x_{\sigma(1)}x_{\sigma(2)} = x_{\sigma(3)}x_{\sigma(4)}
   \]

3. **Using the equation \(x_1x_2 + x_3x_4 = x_1x_4 + x_2x_3\):**
   \[
   (x_1 - x_4)(x_2 - x_3) = 0
   \]
   This implies either \(x_1 = x_4\) or \(x_2 = x_3\).

4. **Assume \(x_1 = x_4\):**
   \[
   x_1x_2 + x_3x_4 = x_1x_4 + x_2x_3
   \]
   Simplifying, we get:
   \[
   (x_1 - x_3)(x_2 - x_4) = 0
   \]
   This implies either \(x_1 = x_3 = x_4\) or \(x_2 = x_3 = x_4\).

5. **At least three variables are equal:**
   Suppose \(x_1 = x_2 = x_3\). Then:
   \[
   x_1 + x_2 + x_3 + x_4 = 3x_1 + x_4
   \]
   Adding all equations, we get:
   \[
   3(x_1 + x_2 + x_3 + x_4) = 3(x_1 + x_2 + x_3 + x_4)^2
   \]
   Simplifying, we get:
   \[
   x_1 + x_2 + x_3 + x_4 \in \{0, 1\}
   \]

6. **Investigate two cases:**
   - Case 1: \(x_1 + x_2 + x_3 + x_4 = 0\)
     \[
     (x_1, x_2, x_3, x_4) = (c, c, c, -3c)
     \]
   - Case 2: \(x_1 + x_2 + x_3 + x_4 = 1\)
     \[
     (x_1, x_2, x_3, x_4) = (c, c, c, 1 - 3c)
     \]

The final answer is \( \boxed{ (x_1, x_2, x_3, x_4) = (c, c, c, -3c) } \) or \((x_1, x_2, x_3, x_4) = (c, c, c, 1 - 3c)\).