In this paper, we consider Whittaker modules for the truncated current Lie algebra where the underlying Lie algebra is of type B2 and the level is one. More precisely, we give the construction of a series of simple Whittaker modules and show that they classify simple Whittaker modules under some conditions. When the Whittaker function is singular, we show that there might exist simple Whittaker modules with non-trivial Whittaker vectors, and we concretely give the expressions of those non-trivial Whittaker vectors. Our results show that the existence of non-trivial Whittaker vectors depends not only on the Whittaker function, but also on the actions of the Casimir elements.
Qiao et al. (Tue,) studied this question.