  The fourth-order ordinary differential equation, defining new transcendents,
is studied. The self-similar solutions of the Kaup-Kupershmidt and
Savada-Kotera equations are shown to be found taking its solutions into
account. Equation studied belongs to the class of fourth-order analogues of the
Painlev\'{e} equations. All the power and non-power asymptotic forms and
expansions near points $z=0$, $z=\infty$ and near arbitrary point $z=z_0$ are
found by means of power geometry methods. The exponential additions to the
solutions of studied equation are also determined.
