Twistor formulation of massless $6D$ infinite spin fields

We construct the massless infinite spin irreducible representations of the six-dimensional Poincare group in 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 full set of equations of motion for the two-twistor fields representing by the totally symmetric $\mathrm{SU}(2)$ rank $2s$ two-twistor spin-tensor and show that they carry out the massless infinite spin representations. The field twistor transform is constructed and the infinite spin fields are found in the space-time formulation with additional spinor coordinate.