${\mathbb{Z}}_2$-graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices

Let $K$ be a field (finite or infinite) of char$(K)\mathrm{\ne }2$ and let $U{T}_n=U{T}_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot$ is the usual product on $U{T}_n$ then with the new product $a\circ b=(1/2)(a\cdot b +b\cdot a)$wehavethat$UT_n$ is a Jordan algebra, denoted by $U{J}_n=U{J}_n(K)$. In this paper, we describe the set of all ${\mathbb{Z}}_2$-graded polynomial identities of $U{J}_2$ with any nontrivial ${\mathbb{Z}}_2$-grading. Moreover, we describe a linear basis for the corresponding relatively free ${\mathbb{Z}}_2$-graded algebra.