  We investigate the problem of similarity to a self-adjoint operator for
$J$-positive Sturm-Liouville operators
$L=\frac{1}{\omega}(-\frac{d^2}{dx^2}+q)$ with $2\pi$-periodic coefficients $q$
and $\omega$. It is shown that if 0 is a critical point of the operator $L$,
then it is a singular critical point. This gives us a new class of $J$-positive
differential operators with the singular critical point 0. Also, we extend the
Beals and Parfenov regularity conditions for the critical point $\infty$ to the
case of operators with periodic coefficients.
