## ABSTRACT

The Painleve property (PP) for partial differential equations (PDE) is invariant under the natural extension of the homographic group and classifications similar to those of ordinary differential equations (ODE) have also been performed for PDEs, in particular second-order first-degree PDEs. Although partially integrable and non-integrable equations, that is the majority of physical equations they retain part of the properties of integrable PDEs and this is why the methods presented apply to both cases as well. Given a Liouville-integrable Hamiltonian system which, in addition, passes the Painleve test, one must try to prove its Painleve integrability by explicitly integrating. Despite the lack of consensus on this definition, a discrete Painleve test has been developed to generate necessary conditions for these properties. Every linear ODE possesses the PP since its general solution depends linearly on the movable constants so, in order to define new functions, one must turn to nonlinear ODEs in a systematic way: first-order algebraic equations, then second-order, etc.