Advice where it can download or buy Livechat software working over Linux, please. So as all popular livechat-clients have been made for use over Windows OS (Liveperson, Kayako etc).

Thanks.

I've used Crafty Syntax before with good results

Provide Support have recently released a Beta Version Of There Linux Chat Console. Which allows you to help your customers and get alerted of new chats, right from your desktop. They offer a 15 day demo which I think you should try. There address is http://www.providesupport.com

I will suggest the Live chat from stardevelop http://livehelp.stardevelop.com/

I will suggest the Live chat from stardevelop http://livehelp.stardevelop.com/

It doesn't work on linux

It doesn't work on linux
I would also recommend StarDevelop. The web interface works regardless of OS.