Gauge symmetries of systems with a finite number of degrees of freedom

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For systems with a finite number of degrees of freedom, it is shown in [arXiv:hep-th/0303014] that first class constraints are Abelianizable if the Faddeev-Popov determinant is not vanishing for some choice of subsidiary constraints. Here, for irreducible first class constraint systems with SO(3) or SO(4) gauge symmetries, including a subset of coordinates in the fundamental representation of the gauge group, we explicitly determine the Abelianizable and non-Abelianizable classes of constraints. For the Abelianizable class, we explicitly solve the constraints to obtain the equivalent set of Abelian first class constraints. We show that for non-Abelianizable constraints there exist residual gauge symmetries which results in confinement-like phenomena.