F. Gungor et al., ON THE INTEGRABILITY PROPERTIES OF VARIABLE-COEFFICIENT KORTEWEG-DE-VRIES EQUATIONS, Canadian journal of physics, 74(9-10), 1996, pp. 676-684
All variable coefficient Korteweg - de Vries (KdV) equations with thre
e-dimensional Lie point symmetry groups are investigated. For such an
equation to have the Painleve property, its coefficients must satisfy
seven independent partial differential equations. All of them an satis
fied only for equations equivalent to the KdV equation itself. However
, most of them are satisfied in all cases. If the symmetry algebra is
either simple, or nilpotent, then the equations have families of singl
e-valued solutions depending on two arbitrary functions of time. Symme
try reduction is used to obtain particular solutions. The reduced ordi
nary differential equations are classified.