  This paper uses the Noether symmetry approach to examine the viable and
stable traversable wormhole solutions in the framework of
$f(\mathcal{R,}\mathcal{T}^{2})$ theory, where $\mathcal{R}$ is the Ricci
scalar and $\mathcal{T}^{2}=\mathcal{T}_{\mu\nu}\mathcal{T}^{\mu\nu}$ is the
self-contraction of stress-energy tensor. For this purpose, we consider a
specific model of this modified theory to obtain exact solutions of the Noether
equations. Further, we formulate the generators of Noether symmetry and first
integrals of motion. We analyze the presence of viable and stable traversable
wormhole solutions corresponding to different redshift functions. In order to
determine whether this theory provides physically viable and stable wormhole
geometry or not, we check the graphical behavior of null energy constraint,
causality condition and adiabatic index for effective stress-energy tensor. It
is found that viable and stable traversable wormhole solutions exist in this
modified theory.
