  Let p be a rational prime. Let F be a totally real number field such that F
is unramified over p and the residue degree of any prime ideal of F dividing p
is 1 or 2. In this paper, we show that the eigenvariety for Res_{F/Q}(GL_2),
constructed by Andreatta-Iovita-Pilloni, is proper at integral weights for
p>=3. We also prove a weaker result for p=2.
