where R(z,w) is a rational function, has a transcendental meromorphic solution, then R is a polynomial of degree at most 2 with respect to w; in other words the differential equation is a Riccati equation, or linear.
Theorem (1920). If an irreducible differential equation
where F is a polynomial, has a transcendental meromorphic solution, then the equation has no movable singularities. Moreover, it can be algebraically reduced either to a Riccati equation or to
where P is a polynomial of degree 3 with respect to w.
Theorem (1941). If an irreducible differential equation
where F is a polynomial, has a transcendental algebroid solution, then it can be algebraically reduced to an equation that has no movable singularities.
A modern account of theorems 1913, 1920 is given in the paper of A. Eremenko(1982)
References
Malmquist, J. (1913), "Sur les fonctions à un nombre fini de branches définies par les équations différentielles du premier ordre", Acta Mathematica, 36 (1): 297–343, doi:10.1007/BF02422385
Malmquist, J. (1941), "Sur les fonchillotions à un nombre fini de branches satisfaisant à une équation différentielle du premier ordre", Acta Mathematica, 74 (1): 175–196, doi:10.1007/BF02392253, MR0005974