Timelike and spacelike Osserman manifolds of signature $(2,2)$ are defined in terms of the characteristic and minimal polynomials of the Jacobi operator (for details see [BBR]). Osserman manifolds with the diagonalizable Jacobi operator are characterized as rank-one symmetric spaces or flat. Geometry of Osserman manifolds with nondiagonalizable Jacobi operator is not yet completely clarified. Some partial answers can be found in [BBR], [BBRa], [BBRb]. In the most general case the Osserman type condition can be expressed in terms of the second order PDE system. In this paper we derive the first order PDE system characterizing Osserman manifolds when the minimal polynomial has a triple zero.