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Single-qubit gate design using oscillatory controls is related to the Rabi problem of rotating a spin. In the classical solution one drives the spin with an oscillatory electromagnetic field orthogonal to a background field. Here, we introduce a new, general method for constructing continuous, oscillatory quantum controls based on Floquet’s theorem. We then derive a family of exact, analytical solutions to the generalized Rabi problem of completely controlling a single-qubit in a fixed background field. We show that for our solutions exact control is only possible if the background field is oblique or zero. We point out that, since no background field need be generated, all single-qubit controls for a Rabi problem are realizable with a simpler, less noisy apparatus in the zero-background-field case.