Where is the Time Machine App?

Why are you asking? It is supposed to run automagically. As soon as you plug in your external drive, your Mac should ask if you want it to be used for Time Machine.

However, if you are trying to inspect the application itself, I would try (but I do not guarantee success) opening a Terminal and typing

<pre>
locate Time\ Machine
</pre>
or
<pre>
locate backupd
</pre>
and inspect the results. If there are too many results to inspect, append " | less" to the command; for example
<pre>
locate backupd | less
</pre>
and then use the spacebar to scroll forwards and 'b' to scroll backwards through the results.

Perhaps Spotlight is an easier way to search but I don't know if Spotlight archives the same locations as locate does, and you said you searched everywhere you could think of.

its in the Applications folder, near the bottom. .../Applications/Time Machine.app

You can just drag the icon back into your dock.

If you have somehow deleted it, you would need to reinstall it off the leopard DVD. Not sure if you can extract it though.

To restore a backup, launch Time Machine. if you somehow completely wiped your main drive and need to totally restore it. Boot from the Leopard Install DVD and follow the menus to "Restore from Time Machine Backup"

I edited my earlier post while you were writing this one.

sounds like you lost the application. I don't know, off the top of my head how to extract TM from the Leopard DVD and install it and only it back onto your computer.

However, if you can get into the Time Machine preferences, and it seems to be backing up your Mac on a regular basis, then you have not in fact deleted or lost it.

When you look in your "Applications" folder, make sure you are looking in the right one. Maybe you know all this already, but sometimes people have an "Applications" folder inside their user directory for software that was installed for that user and only that user to use . If you look at the top level of your Mac hard drive, in what is commonly refered to as the "root" folder, you should see an Applications folder in that. thats where TM is supposed to be.