  We construct an exact solution to the revised small-$x$ orbital angular
momentum (OAM) evolution equations derived recently, based on an earlier work.
These equations are derived in the double logarithmic approximation (summing
powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling constant
and $x$ the Bjorken $x$ variable) and the large-$N_c$ limit, with $N_c$ the
number of quark colors. From our solution, we extract the small-$x$,
large-$N_c$ expressions of the quark and gluon OAM distributions. Additionally,
we determine the large-$N_c$ small-$x$ asymptotics of the OAM distributions to
be
  \begin{align}
  \notag
  L_{q+\bar{q}}(x,Q^2) \sim L_G(x,Q^2) \sim \Delta \Sigma (x,Q^2) \sim \Delta
G(x,Q^2) \sim \left(\frac{1}{x} \right)^{\alpha_h},
  \end{align} with the intercept $\alpha_h$ the same as obtained in the
small-$x$ helicity evolution, which can be approximated as $\alpha_h \approx
3.66074 \sqrt{\frac{\alpha_s N_c}{2\pi}}$. This result is in complete agreement
with the literature. Additionally, we calculate the ratio of the quark and
gluon OAM distributions to the flavor-singlet quark and gluon helicity parton
distribution functions respectively in the small-$x$ region.
