We study the Ornstein-Uhlenbeck-Kac (OU-K) process, where instead of a Brownian motion, a finite-speed Kac process defines the driving process. The OU-K process can be analyzed through a rescaled, time-inhomogeneous Kac process, characterized by a switching rate from the Kac part, and a convergence rate from the Ornstein-Uhlenbeck (OU) part. The law of this inhomogeneous Kac process describes the solution of a wave equation, the Euler-Poisson-Darboux (EPD) equation. For Kac switching rates being multiples of the OU convergence rate, we can explicitly derive this EPD solution via the sum of (the derivative of) two Legendre polynomials, restricted between two Dirac points representing the wave fronts. Using the framework of piecewise-deterministic Markov processes (PDMP), we describe the conditional velocity fields of the OU-K process via our polynomiallaw- formula. This allows us to apply the OU-K process in a generative flow model. Since the OU-K velocity fields prove to be bounded by the finite Kac speed, the usual time truncations of diffusion models become unnecessary. We demonstrate advantages over diffusion and flow matching models on singular data sets and on few-step CIFAR generation. Here, the ability to choose low speeds for the OU-K process proves especially beneficial, yielding almost straight-line trajectories of the flow. Joint work with Richard Duong and Janis Auffenberg.