  In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with
data at infinity. We first prove the existence and uniqueness of it. Then we
show that its symmetric solutions are actually half-space solutions. Our
results are needed when classifying the half-space
$(2k-\frac{1}{2})$-homogeneous solutions to the thin obstacle problems in
$\mathbb{R}^3$. It is a generalization of one part of Savin-Yu's work
\cite{savin2021halfspace} on classifying the half-space
$\frac{7}{2}$-homogeneous solutions.
