  It has recently been demonstrated that asymptotically flat neutral reflecting
stars are characterized by an intriguing no-hair property. In particular, it
has been proved that these {\it horizonless} compact objects cannot support
spatially regular {\it static} matter configurations made of scalar (spin-0)
fields, vector (spin-1) fields, and tensor (spin-2) fields. In the present
paper we shall explicitly prove that spherically symmetric compact reflecting
stars can support {\it stationary} (rather than static) bound-state massive
scalar fields in their exterior spacetime regions. To this end, we solve
analytically the Klein-Gordon wave equation for a linearized scalar field of
mass $\mu$ and proper frequency $\omega$ in the curved background of a
spherically symmetric compact reflecting star of mass $M$ and radius
$R_{\text{s}}$. It is proved that the regime of existence of these stationary
composed star-field configurations is characterized by the simple inequalities
$1-2M/R_{\text{s}}<(\omega/\mu)^2<1$. Interestingly, in the regime
$M/R_{\text{s}}\ll1$ of weakly self-gravitating stars we derive a remarkably
compact {\it analytical} formula for the discrete spectrum
$\{\omega(M,R_{\text{s}},\mu)\}^{n=\infty}_{n=1}$ of resonant oscillation
frequencies which characterize the stationary composed
compact-reflecting-star-linearized-massive-scalar-field configurations.
Finally, we verify the accuracy of the analytically derived resonance formula
of the composed star-field configurations with direct numerical computations.
