  The direct linearization structure is presented of a "mild" but significant
generalization of the lattice BSQ system. Some of the equations in this system
were recently discovered in [J. Hietarinta, J. Phys {\bf A}: Math. Theor. {\bf
44} (2011) 165204] through a search of a class of three-component systems
obeying the property of multidimensional consistency. We show that all the
novel equations arising in this class follow from one and the same underlying
structure. Lax pairs for these systems are derived and explicit expressions for
the $N$-soliton solutions are obtained from the given structure.
