3 × 3 matrices with a flat portion on the boundary of the numerical range

The set of unitarily irreducible complex 3 × 3 matrices whose numerical range has a flat portion is explicitly described. It is proved that the set is connected, and real analytic properties of the set are studied. Connectedness of the sets of unitarily irreducible 3 × 3 matrices having two other possible shapes of their numerical ranges is proved as well.