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As noticed by explained in J.C. Ottem's in a previous comment (now deleted)answer, the generic Enriques surface contains no smooth rational curves at all.

However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves, and also infinitely many of them (see this paper by Cossec and Dolgachev). The maximal number of disjoint $(-2)$ curves on $X$ is eight, and Enriques surfaces with eight disjoint $(-2)$-curves are classified in the article

Mendes Lopes, Margarida; Pardini, Rita

Enriques surfaces with eight nodes

Math. Z. 241 (2002), no. 4, 673–683.

The authors first show that, setting $C_1, \dots,C_8$ to be the exceptional $(-2)$-curves of $X$, the divisor $C_1+\dots+C_8$ is divisible by $2$ in the Picard group of $X$, or equivalently there exists a double cover $\widetilde{X} \to X$ branched exactly over them.

The main theorem then states that an Enriques surface with eight disjoint $(-2)$-curves is isomorphic to $X=D_1\times D_2/G$, where $D_1,D_2$ are elliptic curves and $G$ is either $\mathbb{Z}_2^2$ or $\mathbb{Z}_2^3$.
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As explained in noticed by J.C. Ottem's in a previous comment (now deleted), the generic Enriques surfaces has Picard group of rank one, then it surface contains no rational curves at all.

However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves, and also infinitely many of them (see Ottem's link)this paper by Cossec and Dolgachev). The maximal number of disjoint $(-2)$ curves on $X$ is eight, and Enriques surfaces with eight disjoint $(-2)$-curves are classified in the paperarticle

Mendes Lopes, Margarida; Pardini, Rita

Enriques surfaces with eight nodes

Math. Z. 241 (2002), no. 4, 673–683.

The authors first show that, setting $C_1, \dots,C_8$ to be the exceptional $(-2)$-curves of $X$, the divisor $C_1+\dots+C_8$ is divisible by $2$ in the Picard group of $X$, or equivalently there exists a double cover $\widetilde{X} \to X$ branched exactly over them.

The main theorem then states that an Enriques surface with eight disjoint $(-2)$-curves is isomorphic to $X=D_1\times D_2/G$, where $D_1,D_2$ are elliptic curves and $G$ is either $\mathbb{Z}_2^2$ or $\mathbb{Z}_2^3$.
show/hide this revision's text 2 added 84 characters in body

As explained in J.C. Ottem's comment, the generic Enriques surfaces has Picard group of rank one, then it contains no rational curves at all.

However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves. (-2)$-curves, and also infinitely many of them (see Ottem's link). The maximal number of disjoint $(-2)$ curves on $X$ is eight, and Enriques surfaces with eight disjoint $(-2)$-curves are classified in the paper

Mendes Lopes, Margarida; Pardini, Rita

Enriques surfaces with eight nodes

Math. Z. 241 (2002), no. 4, 673–683.

The authors first show that, setting $C_1, \dots,C_8$ to be the exceptional $(-2)$-curves of $X$, the divisor $C_1+\dots+C_8$ is divisible by $2$ in the Picard group of $X$, or equivalently there exists a double cover $\widetilde{X} \to X$ branched exactly over them.

The main theorem then states that an Enriques surface with eight disjoint $(-2)$-curves is isomorphic to $X=D_1\times D_2/G$, where $D_1,D_2$ are elliptic curves and $G$ is either $\mathbb{Z}_2^2$ or $\mathbb{Z}_2^3$.
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