On integrability of a $q$-oscillator lattice with a $B$-boundary

In this paper we propose a method of construction of a double layer-to-layer auxiliary transfer matrix defined on a half-plane with a boundary. The transfer matrix obtained has the following features: - It produces a complete set of integrals of motion, - Its ingredients can be seen as an auxiliary problem for $3d$ Kuniba-Okado reflection matrix, - The model obtained has no quantum group interpretation.