  Let $\mathbb{F}$ be a field of characteristic $p$, and let $UT_n(\mathbb{F})$
be the algebra of $n \times n$ upper triangular matrices over $\mathbb{F}$ with
an involution of the first kind. In this paper we describe: the set of all
$*$-central polynomials for $UT_n(\mathbb{F})$ when $n\geq 3$ and $p\neq 2$ ;
the set of all $*$-polynomial identities for $UT_3(\mathbb{F})$ when
$\mathbb{F}$ is infinite and $p>2$.
