With many encoders, it's possible to stream to Facebook and Stream Monkey at the same time. You will need to send two streams out of your encoder. One of them will be an rtmp stream sending to Stream Monkey, the other will be sent directly to your Facebook account. 

You want to be sure to create a second output (if your encoder allows it) for Facebook. For example, in Wirecast Studio, you would go to output settings, then the add button at the bottom of the left of the screen to add an output:

In Wirecast, it will let you choose to log in to your Facebook account, and stream directly to them. Be sure that both inputs are checked on the left side of the stream in order to send to both at the same time. 

If you're using an encoder that doesn't have a specified Facebook output, you can choose to send an RTMP stream, and get your stream key (name) and server url (address) from your Facebook account. You will put that information into your encoder in the new output. 

If your encoder is not able to send more than one output (like OBS, or the Matrox Monarch), you can use a second encoder to send another signal. OBS is a free software encoder that can be downloaded onto a computer from the internet, and is a great option for a second encoder. 

Keep in mind that any time you are running more than one software, or sending more than one output, you'll need more bandwidth and processing power. Run tests ahead of time, and keep an eye on your connection and CPU to make sure they are not overworked. 

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