  Light-cone gauge formulation of relativistic dynamics of a continuous-spin
field propagating in the flat space is developed. Cubic interaction vertices of
continuous-spin massless fields and totally symmetric arbitrary spin massive
fields are studied. We consider parity invariant cubic vertices that involve
one continuous-spin massless field and two arbitrary spin massive fields and
parity invariant cubic vertices that involve two continuous-spin massless
fields and one arbitrary spin massive field. We construct the complete list of
such vertices explicitly. Also we demonstrate that there are no cubic vertices
describing consistent interaction of continuous-spin massless fields with
arbitrary spin massless fields.
