We consider the family of selfadjoint operators of the form $H + tV$ in a Hilbert space. Here $H$ and $V$ are selfadjoint operators and $t$ is a complex parameter. It is assumed that the range of $V$ is a generating for $H$. We discuss when the set of $t$ in $[0,1$] ,such that a fixed real number is the eigenvalue of the operators $H+tV$ , should be of the Lebesque measure $0$.In particular in the case of nonnegative $V$ it is true and show by explicit counterexamples that the nonnegativity assumption cannot be omitted. The talk is based on the common work with F.Gesztesy and R.Nichols. Journal of Mathematical Analysis and Applications. 428 (2015) pp. 295-305