DELAY-DIFFERENTIAL EQUATIONS AND THE PAINLEVE TRANSCENDENTS

We apply the recently proposed integrability criterion for differential-difference systems (that blends the classical Painleve analysis with singularity confinement for discrete systems) to a class of first-order differential-delay equations. Our analysis singles out the family of bi-Riccati equations, as integrability candidates. Among these equations that pass the test some are integrable in a straightforward way (usually by reduction to a standard Riccati equation for some transformed variable) while the remaining ones define new hysterodifferential forms of the Painleve transcendental equations.