I would like to know if I can integrate Lithium with other systems like Avaya and / or eGain so I have one universal queue with intellegent routing base, so any agent can handle any transaction on any of platforms ?
That sounds like a dream set-up @GeorgeAttia!
I suggest you start by looking at Lithium Social Web (LSW) so you can funnel all social support channels into the one pipe. Integrating other channels like phone support and LiveChat would require significant customisation.
About a year ago we did an integration with our social media tool. Our SoMe provider (not lithium) isn't as big as Avaya, and it took some custom development, but i'm happy with the result.