EE Don't offer a static public IP address option on their home broadband.
If you really really need one then you will need to go with another provider: Zen, A&A etc.
The alternatives if you wish to remain with EE for home broadband are:
1. Invest in a DDNS solution ie. both your router and domain registrar will need to support this. Draytek with their entry-level routers support DDNS and they offer a free DrayDDNS service which would be sufficient for home usage. However, for this to work reliably (if you want to use a domain name for your home server that isn't the one provided by the DDNS provider) you will need to ensure your domain registrar supports the frequent update of DNS records. 123-reg, for example use a fixed expiry time of 4 hours which isn't much use with DDNS which really needs an expiry time of sub 10 minutes.
2. Go with an overlay solution such as A&A's L2TP-VPN which will give you a static IP address, but will require your router to be configured to maintain an outbound L2TP-VPN connection. (Aside: this approach currently also works over EE's mobile broadband, so could be useful if you want a backup in case the fixed broadband service breaks...)
