  We construct families of bispectral difference operators of the form a(n)T +
b(n) + c(n) T^{-1} where T is the shift operator. They are obtained as discrete
Darboux transformations from appropriate extensions of Jacobi operators. We
conjecture that along with operators previously constructed by Grunbaum, Haine,
Horozov, and Iliev they exhaust all bispectral regular (i.e. a(n)c(n) \neq 0,
for all integer n) operators of the form above.
