We consider the random βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg-transformation βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg, defined on βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg], that generates all possible expansions of the form βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg, whereβKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg}. This transformation was introduced in [3–5], where two naturalinvariant ergodic measures were found. The first is the unique measure ofmaximal entropy, and the second is a measure of the form βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg, with βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβgthe Bernoulli βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg product measure and βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg is a measure equivalent to theLebesgue measure. In this paper, we give an uncountable family of βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg-invariantexact βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg-measures for a certain collection of algebraic βKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg’s. The construction of theseβKβ{0,1}N×[0,⌊β⌋]β-1]x=∑i=0∞aiβiai∈{0,1,…,⌊β⌋}mp×μβmp(p,1-p)μβKβgβg-measures is explicit and the corresponding potentials are not locally constant.