Given is a natural number $n \geq 3$. Solve the system of equations:
$\[
\begin{cases}
\tan (x_1) + 3 \cot (x_1) &= 2 \tan (x_2) \\
\tan (x_2) + 3 \cot (x_2) &= 2 \tan (x_3) \\
& \dots \\
\tan (x_n) + 3 \cot (x_n) &= 2 \tan (x_1) \\
\end{cases}
\]$
To solve the given system of equations, we will analyze the function and the behavior of the tangent and cotangent functions.

Given the system of equations:
\[
\begin{cases}
\tan (x_1) + 3 \cot (x_1) &= 2 \tan (x_2) \\
\tan (x_2) + 3 \cot (x_2) &= 2 \tan (x_3) \\
& \vdots \\
\tan (x_n) + 3 \cot (x_n) &= 2 \tan (x_1)
\end{cases}
\]

1. **Rewrite the equations in terms of a single variable:**
   Let \( y_k = \tan(x_k) \). Then the system becomes:
   \[
   \begin{cases}
   y_1 + \frac{3}{y_1} &= 2 y_2 \\
   y_2 + \frac{3}{y_2} &= 2 y_3 \\
   & \vdots \\
   y_n + \frac{3}{y_n} &= 2 y_1
   \end{cases}
   \]

2. **Analyze the function \( f(z) = \frac{z}{2} + \frac{3}{2z} \):**
   We need to find the fixed points of this function. A fixed point \( z \) satisfies:
   \[
   z = \frac{z}{2} + \frac{3}{2z}
   \]
   Multiplying both sides by \( 2z \) to clear the fraction:
   \[
   2z^2 = z^2 + 3 \implies z^2 = 3 \implies z = \pm \sqrt{3}
   \]

3. **Behavior of the function \( f(z) \):**
   - For \( z > 0 \), \( f(z) \) has a unique positive fixed point at \( z = \sqrt{3} \).
   - For \( z < 0 \), \( f(z) \) has a unique negative fixed point at \( z = -\sqrt{3} \).

4. **Convergence to fixed points:**
   - If \( y_1 \neq \sqrt{3} \), then \( y_1 > \sqrt{3} \) and \( y_n > \sqrt{3} \) implies \( \sqrt{3} < y_{n+1} < y_n \), showing that all positive seeds have orbits converging to \( \sqrt{3} \).
   - By symmetry, all negative seeds have orbits converging to \( -\sqrt{3} \).

5. **Conclusion for the system:**
   Since all \( y_k \) converge to either \( \sqrt{3} \) or \( -\sqrt{3} \), we have:
   \[
   \tan(x_1) = \tan(x_2) = \cdots = \tan(x_n) = \pm \sqrt{3}
   \]
   Therefore, the solutions for \( x_k \) are:
   \[
   (x_1, x_2, \ldots, x_n) \in \pi \mathbb{Z}^n \pm \left( \frac{\pi}{3}, \frac{\pi}{3}, \ldots, \frac{\pi}{3} \right)
   \]

The final answer is \( \boxed{ (x_1, x_2, \ldots, x_n) \in \pi \mathbb{Z}^n \pm \left( \frac{\pi}{3}, \frac{\pi}{3}, \ldots, \frac{\pi}{3} \right) } \).