  Let $H$ be a complex Hilbert space of dimension not less than $3$ and let
${\mathcal C}$ be a conjugacy class of compact self-adjoint operators on $H$.
Suppose that the dimension of the kernels of operators from ${\mathcal C}$ not
less than the dimension of their ranges. In the case when ${\mathcal C}$ is
formed by operators of finite rank $k$ and $\dim H=2k$, we assume that $k\ge
4$. We show that every bijective transformation of C preserving the
commutativity in both directions is induced by a unitary or anti-unitary
operator up to a permutation of eigenspaces of the same dimension.
