Demushkin's Theorem in codimension one

Demushkin's Theorem says that any two toric structures on an affine variety X are conjugate in the automorphism group of X. We provide the following extension: Let an (n–1)-dimensional torus T act effectively on an n-dimensional affine toric variety X. Then T is conjugate in the automorphism group of X to a subtorus of the big torus of X. Mathematics Subject Classification: 13A50, 14L30, 14M25, 14R20.