  Let $\mathbb{T}$ be the three dimensional torus acting on $\mathbb{P}^{3}$
and $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ be the fixed locus of the
corresponding action on the moduli space of rank $2$ framed instanton sheaves
on $\mathbb{P}^{3}.$ In this work, we prove that
$\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ consist only of non locally-free
instanton sheaves whose double dual is the trivial bundle
$\mathcal{O}_{\mathbb{P}^{3}}^{\oplus 2}$. Moreover, We relate these instantons
to Pandharipande-Thomas stable pairs and give a classification of their
support. This allows to compute a lower bound on the number of components of
$\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c).$
