  There are several approaches to formulate gauge-invariant models for massive
integer-spin field in $d$ dimensions including the following: (i) in terms of
symmetric tensor fields $\phi_{\mu_1 \dots \mu_k} $, with $k = s, s-1, \dots ,
0$, restricted to be double traceless for $k\geq 4$; and (ii) in terms of a
quartet of $traceful$ symmetric tensor fields $\psi_{\mu_1 \dots \mu_k} $, of
rank $k=s,s-1,s-2, s-3$. We demonstrate that these formulations in Minkowski
space ${\mathbb M}^d$ are equivalent to the gauge-invariant theory for a
massive integer-spin field proposed in 1989 by Pashnev. We also make use of the
Klishevich-Zinoviev theory in ${\mathbb M}^d$ to derive a generalisation of the
Singh-Hagen model for a massive integer-spin field in $d>4 $ dimensions.
