Vibration of a spinning, cyclic symmetric rotor supported by flexible bearings and housing is governed by a set of ordinary differential equations with periodic coefficients. As a result, analytical solutions of such systems are generally not available. This paper is to prove that closed-form solutions are available for such systems if the following two conditions are met. First, the rotor has a rigid hub and the rest of the rotor is flexible. Second, elastic mode shapes of the rotor's flexible part only present axial displacement. Under these two conditions, the periodic coefficients will only appear between repeated modes of the spinning rotor and vibration modes of the stationary housing. This unique structure enables a coordinate transformation to convert the governing ordinary differential equations with periodic coefficients into a set of ordinary differential equations with constant coefficients, whose closed-form solution is readily available. Moreover, the coordinate transformation can be derived explicitly. Finally, we demonstrate the closed-form solution through a benchmark numerical model that consists of a spinning rotor, a stationary housing, and two elastic bearings. In particular, the rotor is a circular disk with four evenly spaced radial slots and a central rigid hub. The housing is a square plate with a central rigid shaft and is fixed at four corners. The two elastic bearings connect the rotor and the housing between the hub and shaft. Numerical results confirm that the original equation of motion with periodic coefficients and the closed-form solutions predict the same vibration response.