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Let E and F be two complex Hilbert spaces. Let S_{E} and S_{F} be the shift operators on vector-valued Hardy space H_{E}^{2} and H_{F}^{2}, respectively. We show that the cyclic multiplicity of S_{E}\oplus S_{F} ^{\ast } equals 1+\dim E. This result is classical when \dim E=\dim F=1 (see J. A. Deddens [ On A \oplus A^\ast, 1972]; Paul Richard Halmos [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos's book [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for S_{E}\oplus S_{F}^{\ast } (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of S_{E}\oplus S_{F}^{\ast } when \dim E<\infty and \dim F=1. [ABSTRACT FROM AUTHOR]