Given, the direction cosines of two lines are $\left(\frac{2}{3}, \frac{2}{3}, \frac{1}{3}\right)$ and $\left(\frac{5}{13}, \frac{12}{13}, 0\right).$

We know that, the direction ratios of the line bisecting the angle between the lines whose direction cosines are $\left(l_1, m_1, n_1\right)$ and $\left(l_2, m_2, n_2\right)$ are proportional to $⟨l_1+l_2, m_1+m_2, n_1+n_2⟩.$

Hence, the required direction ratios are proportional to $⟨\frac{2}{3}+\frac{5}{13}, \frac{2}{3}+\frac{12}{13}, \frac{1}{3}+0⟩=⟨\frac{41}{39}, \frac{62}{39}, \frac{13}{39}⟩.$

Thus, the direction ratios are $⟨41, 62, 13⟩.$