We present here three new exact solutions of Brans-Dicke theory for a stati
onary geometry with cylindrical symmetry in the presence of matter in rigid
rotation with T-mu (mu) not equal 0. All the solutions have eternal closed
timelike curves in some region of space-time which has a size that depends
on omega. Moreover, two of them do not go over a solution of general relat
ivity in the limit omega --> infinity.