2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/114341The 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 Painlevé 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.31 pages, 9 figuresExactly Solvable and Integrable Systemstext