With an Enterprise plan, you may remove all Livestream branding and links to create a completely white-labeled embedded player.
Click on the event from your dashboard, then click the Editor icon.
Choose the 'Privacy' tab towards the bottom of the menu.
Under the 'Embedding' section, check 'Hide Livestream branding on embeds.' Then save your settings
Now embed your event on your website.
Whether you are live or not, your embed will appear with no Livestream logos or links to livestream.com. Note that this also disables any built-in social sharing and chat features, since those are all include links to livestream.com.