  In this paper, we explore the conformal structure of singularities arising
from varying fundamental constants using the method of Penrose diagrams. We
employ a specific type of bimetric model featuring two different metrics. One
metric describes the causal structure for matter, while the other characterizes
the causal structure for gravitational interactions, which is related to
variations in fundamental constants such as the gravitational constant and the
speed of light. For this reason, we focused on the gravitational metric to
calculate the conformal transformation and compose Penrose diagrams for the
singularities arising from the varying fundamental constants. We have shown
that, in one case, the parameter such as the scale factor, the density and the
pressure resemble those of the finite scale factor singularity (FSF). Despite
singularity appears in constant conformal time in our case and in the case of
FSF the Misner-Sharp horizon looks different. Our another case is similar to
sudden future singularity (SFS), but there are differences in the conformal
structures. We have also shown that in our cases the behavior of Misner-Sharp
horizon strongly depends on initial conditions. The last analytical solution
which we introduced is identical to conformal structure of the standard exotic
singularity for the matter.
