A recent work has shown that using conformal mapping can lead to exact values of the degenerate scales in plane elasticity. We elaborate on this work by introducing some algebraic tools when this conformal mapping is a rational fraction transforming the outside of the unit circle into the outside of the considered domain. Using these tools, new cases are solved including shortened hypotrochoid, arc of circle, new approximates of equilateral triangle and square or symmetric Joukowski profiles. Another method makes it possible to obtain the degenerate scales for plane elasticity from the degenerate scale for Laplace's equation for some multiply connected sets: the cases of segments on a line or of arcs of circle with a n-fold symmetry. In these last cases, the exact values of the degenerate scales are obtained when the degenerate scale for the Laplace problem is known.