  Massive spin-2 particles has been a subject of great interest in current
research. If the graviton has a small mass, the gravitational force at large
distances decreases more rapidly, which could contribute to explain the
accelerated expansion of the universe. The massive spin-2 particles are
commonly described by the known Fierz-Pauli action which is formulated in terms
of a symmetric tensor $h_{\mu\nu}=h_{\nu\mu}$. However, the Fierz-Pauli theory
is not the only possible description of massive spin-2 particles via a rank-2
tensor. There are other two families of models $\mathcal{L}(a_1)$ and
$\mathcal{L}_{nFP}(c)$, where $a_1$ and $c$ are real arbitrary parameters,
which describe massive particles of spin-2 in the flat space via a nonsymmetric
tensor $e_{\mu\nu}\neq e_{\nu\mu}$. In the present work we derive Lagrangian
constraints stemming from $\mathcal{L}(a_1)$ and $\mathcal{L}_{nFP}(c)$ in
curved backgrounds with nonminimal couplings which are analytic functions of
$m^2$. We show that the constraints lead to a correct counting of degrees of
freedom if nonminimal terms are included with fine tuned coefficients and the
background space is of the Einstein type, very much like the Fierz-Pauli case.
We also examine the existence of local symmetries.
