Integrable Systems of Neumann Type

We construct families of integrable systems that interpolate between -dimensional harmonic oscillators and Neumann systems. This is achieved by studying a family of integrable systems generated by the Casimir functions of the Lie algebra of real skew-symmetric matrices and a certain deformation thereof. Involution is proved directly, since the standard involution theorems do not apply to these families. It is also shown that the integrals are independent.