In this paper, the authors consider Lame operators of the form \[ L=-\frac{d^{2}}{dx^{2}}+V(x) \] with a complex valued periodic potential \[ V(x)=m(m+1)\omega ^{2}\rho (\omega x+z_{0}), \] where \(m\in \mathbb{N}\), \(\omega \) is any halfperiod of \(\rho (x)\). The only assumption on \( z_{0}\in \mathbb{C}\) is that the corresponding potential is non-singular, i.e., the potential is real, periodic and regular.
In the fist part of the paper, it is proved that the opened gaps are precisely the first \(m\) ones. In the second part of the paper, the authors study genuinely complex instances of the Lame operator, concentrating on the simplest case \(m=1,\) when the spectrum consists of two regular analytic arcs, ones of which extends to infinity. Finally, the spectrum in the rhombic Lame case with \(m=2\) is briefly discussed.