  The peculiarity of the Eckart potential problem on ${\bf H}^2_+$ (the upper
sheet of the two-sheeted two-dimensional hyperboloid), to preserve the
$(2l+1)$-fold degeneracy of the states typical for the geodesic motion there,
is usually explained in casting the respective Hamiltonian in terms of the
Casimir invariant of an so(2,1) algebra, referred to as potential algebra. In
general, there are many possible similarity transformations of the symmetry
algebras of the free motions on curved surfaces towards potential algebras,
which are not all necessarily unitary. In the literature, a transformation of
the symmetry algebra of the geodesic motion on ${\bf H}^2_+$ towards the
potential algebra of Eckart's Hamiltonian has been constructed for the prime
purpose to prove that the Eckart interaction belongs to the class of Natanzon
potentials. We here take a different path and search for a transformation which
connects the $(2l+1)$ dimensional representation space of the pseudo-rotational
so(2,1) algebra, spanned by the rank-$l$ pseudo-spherical harmonics, to the
representation space of equal dimension of the potential algebra and find a
transformation of the scaling type. Our case is that in so doing one is
producing a deformed isometry copy to ${\bf H}^2_+$ such that the free motion
on the copy is equivalent to a motion on ${\bf H}^2_+$, perturbed by a $\coth$
interaction. In this way, we link the so(2,1) potential algebra concept of the
Eckart Hamiltonian to a subtle type of pseudo-rotational symmetry breaking
through ${\bf H}^2_+$ metric deformation.
