Why can I not move purchased songs onto my I pod if the computer is authorized?????

Once you downloaded the song in your computer you can trasfer them to your ipad only through a USB connection and using iTunes.

Further, songs copy can start only if your ipad is authorized. You can authorize it using iTunes.

Regards, Francesco

Do you mean sync from your iTunes library on your computer?

If so:

What exactly happens when you try to sync?

Does iTunes see the iPod?

Was the iPod previous synced to another iTunes library/computer?

Have you successfully synced from this iTunes library/computer before?

If so have you done anything like update iTunes on the computer since it last successfully synced?

Do the songs play in iTunes?

Does any media now sync to the iPod?

Do you have the right boxes checked to sync?

iTunes: Syncing media content to iOS devices and iPod

Try syncing using the manual method

Managing content manually on iPhone, iPad, and iPod