To solve the equation \( \frac{x - 1}{3} = \frac{2 - x}{-2x - \frac{1}{3}} \), we will start by simplifying the expression.

1. **Start with the original equation**:
\[
\frac{x - 1}{3} = \frac{2 - x}{-2x - \frac{1}{3}}
\]

2. **Simplify the right side**: The denominator \(-2x - \frac{1}{3}\) can be expressed with a common denominator:
\[
-2x - \frac{1}{3} = \frac{-6x - 1}{3}
\]
Thus, the right side becomes:
\[
\frac{2 - x}{\frac{-6x - 1}{3}} = (2 - x) \cdot \frac{3}{-6x - 1} = \frac{3(2 - x)}{-6x - 1}
\]

3. **Now rewrite the original equation**:
\[
\frac{x - 1}{3} = \frac{3(2 - x)}{-6x - 1}
\]

4. **Cross-multiply** to eliminate the fractions:
\[
(x - 1)(-6x - 1) = 3(2 - x) \cdot 3
\]

5. **Expand both sides**:
- Left side:
\[
-6x^2 - x + 6x + 1 = -6x^2 + 5x + 1
\]
- Right side:
\[
9(2 - x) = 18 - 9x
\]

6. **Set the equation**:
\[
-6x^2 + 5x + 1 = 18 - 9x
\]

7. **Rearranging everything to one side gives**:
\[
-6x^2 + 5x + 1 + 9x - 18 = 0
\]
Which simplifies to:
\[
-6x^2 + 14x - 17 = 0
\]

8. **Multiply through by -1 to simplify**:
\[
6x^2 - 14x + 17 = 0
\]

9. **Use the quadratic formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 6 \cdot 17}}{2 \cdot 6}
\]
Calculate the discriminant:
\[
196 - 408 = -212 \text{ (which is negative)}
\]

Since the discriminant is negative, there are no real solutions for this equation. The solutions would be complex.

10. **Finding the complex solutions**:
\[
x = \frac{14 \pm \sqrt{-212}}{12} = \frac{14 \pm \sqrt{212}i}{12} = \frac{14 \pm 2\sqrt{53}i}{12} = \frac{7 \pm \sqrt{53}i}{6}
\]

So the solutions are:
\[
x = \frac{7 + \sqrt{53}i}{6} \quad \text{and} \quad x = \frac{7 - \sqrt{53}i}{6}
\]