  We study a bifurcation problem for a stationary nonlinear Schr\"{o}dinger
equations with a pure power nonlinearity of exponent $p>1$ on the product space
$(x, y)\in \mathbb{R} \times \mathbb{T}$. The equation has an explicit
solution, called line soliton, which is independent of the transverse direction
$y$. We construct a bifurcation branch. from the line soliton. Such a result
has already been known when $p \geq 2$, and our main result extends the
construction of the branch to the full range of nonlinear exponents $p > 1$.
Since our condition on the nonlinearity is weak, we need to elaborate more when
we apply the Crandall-Rabinowitz Theorem, taking full advantage of the crucial
qualitative properties of the line soliton.
