  We construct massless infinite spin irreducible representations of the
six-dimensional Poincar\'{e} group in the space of fields depending on twistor
variables. It is shown that the massless infinite spin representation is
realized on the two-twistor fields. We present a full set of equations of
motion for two-twistor fields represented by the totally symmetric
$\mathrm{SU}(2)$ rank $2s$ two-twistor spin-tensor and show that they carry
massless infinite spin representations. A field twistor transform is
constructed and infinite spin fields are found in the space-time formulation
with an additional spinor coordinate.
