Theorem (1913). If the differential equation

${\displaystyle {\frac {dw}{dz}}=R(z,w)}$ 

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

${\displaystyle F\left({\frac {dw}{dz}},w,z\right)=0}$ 

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

${\displaystyle \left({\frac {dw}{dz}}\right)^{2}=a(z)P(z,w),}$ 

where P is a polynomial of degree 3 with respect to w.

Theorem (1941). If an irreducible differential equation

${\displaystyle F\left({\frac {dw}{dz}},w,z\right)=0,}$ 

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)

• 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. (1920), "Sur les fonctions à un nombre fini de branches satisfaisant à une équation différentielle du premier ordre" (PDF), Acta Mathematica, 42 (1): 317–325, doi:10.1007/BF02404413
• Malmquist, J. (1941), "Sur les fonctions à un nombre fini de branches satisfaisant à une équation différentielle du premier ordre", Acta Mathematica, 74 (1): 175–196, doi:10.1007/BF02392253, MR 0005974
• Eremenko, A. (1982), "Meromorphic solutions of algebraic differential equations", Russian Mathematical Surveys, 37 (4): 61–95, Bibcode:1982RuMaS..37...61E, doi:10.1070/rm1982v037n04abeh003967, MR 0667974, S2CID 250879409