We consider a family S=S(a) of 2-valued transformations of special form on
the segment [0,1] with measure $\mu=\int p(x) d\lambda$, which is absolutely
continuous with respect to the Lebesgue measure $\lambda$. We endow S with a
set of weight functions $\alpha=\{\alpha_1(x),\alpha_2(x)\}$ and find a
criterion of measure invariance under the transformation. This criterion
relates the three parameters $a$, $p$, $\alpha$ to each other.