In this paper, we study an ODE of the form$$b_0 u^{(4)} + b_1 u'' + b_2 u + b_3 u^3 + b_4 u^5 = 0, \ ' = \frac{d}{d z},$$ which includes, as a special case, the stationary case of the cubic-quintic Swift-Hohenberg equation. Based on Nevanlinna theory and PainlevĂ© analysis, we first show that all its meromorphic solutions are elliptic or degenerate elliptic. Then we obtain them all explicitly by the method introduced in [7].