  Spinor description for the curvatures of $D=5$ Yang-Mills, Rarita-Schwinger
and gravitational fields is elaborated. Restrictions imposed on the curvature
spinors by the dynamical equations and Bianchi identities are analyzed. In the
absence of sources symmetric curvature spinors with $2s$ indices obey
first-order equations that in the linearized limit reduce to Dirac-type
equations for massless free fields. These equations allow for a higher-spin
generalization similarly to $4d$ case. Their solution in the form of the
integral over Lorentz-harmonic variables parametrizing coset manifold
$SO(1,4)/(SO(1,1)\times ISO(3))$ isomorphic to the three-sphere is considered.
Superparticle model that contains such Lorentz harmonics as dynamical
variables, as well as harmonics parametrizing the two-sphere $SU(2)/U(1)$ is
proposed. The states in its spectrum are given by the functions on $S^3$ that
upon integrating over the Lorentz harmonics reproduce on-shell symmetric
curvature spinors for various massless supermultiplets of $D=5$ space-time
supersymmetry.
