Let n≥2 be an integer. To each irreducible representation σ of O(1), an O(1)-Kepler problem in dimension n is constructed and analyzed. This system is super integrable and when n=2 it is equivalent to a generalized MICZ (McIntosh-Cisneros-Zwanziger)-Kepler problem in dimension 2. The dynamical symmetry group of this system is Sp˜(2n,R) with the Hilbert space of bound states H(σ) being the unitary highest weight representation of Sp˜(2n,R) with highest weight which occurs at the rightmost nontrivial reduction point in the Enright–Howe–Wallach classification diagram for the unitary highest weight modules. (Here |σ|=0 or 1 depending on whether σ is trivial or not.) Furthermore, it is shown that the correspondence σ↔H(σ) is the theta correspondence for dual pair (O(1),Sp(2n,R))⊆Sp(2n,R).