  We study the entanglement entropy of a one-parameter family of exactly
solvable gravities in the 2-dimensional asymptotically-flat space. The islands
and Page curves of eternal, evaporating and bath-removed black holes are
investigated. The different theories in this parameter class are identified
through field redefinitions which leave the island invariant. The Page
transition is found to occur at the first a third of the black hole life time
in the evaporating case for this family of solutions. In addition, we consider
gluing the equilibrium black hole and the evaporating one along a null
trajectory and study the effect of gluing on the islands and Page curves. In
the glued space, the island jumps across two different geometries at a certain
retarded time. As a result, the Page transition is stretched and split into two
separate ones -- the first transition happens when the net entropy generation
stops and the second one occurs as the early radiation effectively starts to
become purified. Finally, we discuss the issues concerning the inconsistent
rates of purification and the paradox related to the state of the radiation.
