  We study multiple eigenvalues of a magnetic Aharonov-Bohm operator with
Dirichlet boundary conditions in a planar domain. In particular, we study the
structure of the set of the couples position of the pole-circulation which keep
fixed the multiplicity of a double eigenvalue of the operator with the pole at
the origin and half-integer circulation. We provide sufficient conditions for
which this set is made of an isolated point. The result confirms and validates
a lot of numerical simulations available in preexisting literature.
