  We construct rotating boson stars and Myers-Perry black holes with scalar
hair (MPBHsSH) as fully non-linear solutions of five dimensional Einstein
gravity minimally coupled to a complex, massive scalar field. The MPBHsSH are,
in general, regular on and outside the horizon, asymptotically flat, and
possess angular momentum in a single rotation plane. They are supported by
rotation and have no static limit. Such hairy BHs may be thought of as bound
states of boson stars and singly spinning, vacuum MPBHs and inherit properties
of both these building blocks. When the horizon area shrinks to zero, the
solutions reduce to (in a single plane) rotating boson stars; but the extremal
limit also yields a zero area horizon, as for singly spinning MPBHs. Similarly
to the case of equal angular momenta, and in contrast to Kerr black holes with
scalar hair, singly spinning MPBHsSH are disconnected from the vacuum black
holes, due to a mass gap. We observe that for the general case, with two
unequal angular momenta, the equilibrium condition for the existence of MPBHsSH
is $w=m_1\Omega_1+m_2\Omega_2$, where $\Omega_i$ are the horizon angular
velocities in the two independent rotation planes and $w,m_i$, $i=1,2$, are the
scalar field's frequency and azimuthal harmonic indices.
