We consider equivalence, stability and integration of quadratic Hamilton–Poisson systems on the semi-Euclidean Lie–Poisson space se(1,1)∗−. The inhomogeneous positive semidefinite systems are classified (up to affine isomorphism); there are 16 normal forms. For each normal form, we compute the symmetry group and determine the Lyapunov stability nature of the equilibria. Explicit expressions for the integral curves of a subclass of the systems are found. Finally, we identify several basic invariants of quadratic Hamilton–Poisson systems.