Difference analogue of the Treibich–Verdier operator

In [1], it was shown that the one-dimensional finite-gap Schrödinger operator can be extended to a second-order difference operator depending on a small parameter and commuting with some difference operator of order $2g+1.$ In this case, if the small parameter tends to zero, then the second-order difference operator becomes a Schrödinger operator. In this paper, we construct such an extension for the finite-gap Treibich–Verdier operator.