ElGamal with elliptic curves
Your answer is in the paper Elliptic curve cryptosystems from Neal Koblitz:


Set up an elliptic curve $E$ over a field $\mathbb{F}_q$ and a point $P$ of order $N$ just the same as for EC-DDH  as system parameters.
You need a public known function $f : m \mapsto P_m$, which maps messages $m$ to points $P_m$ on $E$. It should be invertible, and one way is to use $m$ in the curve's equation as $x$ and calculate the according $y$.
Choose a secret key $x \in_R [1,N-1]$ randomly, publish the point $Y=x P$ as public key.
Encryption: choose random $k\in_R [1,N-1]$ , then calculate $C=kP$ and $C'=kY$ and calculate $P_m = f(m)$. The ciphertext is the tuple $(C, C'+P_m)$.
Decryption: From a ciphertext $(C,D)$ calculate $C' = xC$, and retrieve the point $P_m$ with $P_m = D-C' = (k(xP)+P_m)-(x(kP))$. Then calculate the message $m$ with $f^{-1}(P_m)$.