Let
E
E
and
F
F
be two complex Hilbert spaces. Let
S
E
S_{E}
and
S
F
S_{F}
be the shift operators on vector-valued Hardy space
H
E
2
H_{E}^{2}
and
H
F
2
H_{F}^{2}
, respectively. We show that the cyclic multiplicity of
S
E
⊕
S
F
∗
S_{E}\oplus S_{F} ^{\ast }
equals
1
+
dim
E
1+\dim E
. This result is classical when
dim
E
=
dim
F
=
1
\dim E=\dim F=1
(see J. A. Deddens [On
A
⊕
A
∗
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
⊕
S
F
∗
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
⊕
S
F
∗
S_{E}\oplus S_{F}^{\ast }
when
dim
E
>
∞
\dim E>\infty
and
dim
F
=
1
\dim F=1
.