To solve the equation \( 2 : \left( x - \frac{1}{3} \right) - \frac{5}{9} = \frac{1}{92} \), let's first clarify it.

The expression \( 2 : \left( x - \frac{1}{3} \right) \) can be interpreted as \( 2 / \left( x - \frac{1}{3} \right) \). Therefore, we can rewrite the equation as:

\[
\frac{2}{x - \frac{1}{3}} - \frac{5}{9} = \frac{1}{92}
\]

Now, let's isolate the term with \( x \):

1. Add \( \frac{5}{9} \) to both sides:

\[
\frac{2}{x - \frac{1}{3}} = \frac{1}{92} + \frac{5}{9}
\]

2. To combine \( \frac{1}{92} \) and \( \frac{5}{9} \), we need a common denominator. The least common multiple of 92 and 9 is 828. We'll convert both fractions.

- For \( \frac{1}{92} \):
\[
\frac{1}{92} = \frac{9}{828}
\]

- For \( \frac{5}{9} \):
\[
\frac{5}{9} = \frac{460}{828}
\]

So, we have:

\[
\frac{9}{828} + \frac{460}{828} = \frac{469}{828}
\]

Now, the equation looks like:

\[
\frac{2}{x - \frac{1}{3}} = \frac{469}{828}
\]

3. Now we can cross-multiply:

\[
2 \cdot 828 = 469 \left( x - \frac{1}{3} \right)
\]

This simplifies to:

\[
1656 = 469 \left( x - \frac{1}{3} \right)
\]

4. Distribute \( 469 \):

\[
1656 = 469x - \frac{469}{3}
\]

5. To eliminate the fraction, multiply everything by 3:

\[
3 \times 1656 = 3 \times 469x - 469
\]

This gives:

\[
4968 = 1407x - 469
\]

6. Add \( 469 \) to both sides:

\[
4968 + 469 = 1407x
\]

\[
5437 = 1407x
\]

7. Now divide both sides by \( 1407 \):

\[
x = \frac{5437}{1407}
\]

Now, we can simplify \( \frac{5437}{1407} \). When calculated, this fraction does not simplify further, so:

\[
x \approx 3.866 (or \frac{5437}{1407})
\]

So, the solution to the equation is

\[
x = \frac{5437}{1407}
\]