ECDSA with common nonce?
You are right. And this had been exploited in the past in the PS3 and Bitcoin wallet events. 

The first half of the signature is the x coordinate of $k \times G$, and $k$ is the ephemeral key that must be unique to each signature generated. If it's static then the following formula would recover the private signing key: 

$k = {{z_1 - z_2} \over {s_1 - s_2}}$ and $d = {{sk-z} \over r}$ where $z_1$ and $z_2$ are the truncated hash bitstring, $s_1$ and $s_2$ are the two corresponding signature component, $r$ being the fixed part in both signatures, and $d$ the private key we hunt. 

For more, Wikipedia has an article covering ECDSA and its security.