Ok... here it is...

Version 4.6.691 of Avast Home edition Downloaded)
OS on my PC is Windows XP.
I use SP2 Firewall; Lavasoft Adaware; Spyboat & Secretmaker.
The Resident scanner is set properly in the AVAST settings. The icon, within a second or so of its appearance in the system tray, vanishes, never to appear again.
After reading your advice to Grandpaj, I disabled the Firewall and the Spywares, making sure my system tray is vacant. Still, the AVAST icon that appeared, as narrated earlier, refused to stay on.

Is there any way, I could get the 'eye' back, please?