Hi Ivan!
I don't work for IPS, so this is just my own experience as a customer. 🙂
PowWeb was actually my first shared hosting service MAAAANY years ago. Seeing them brought back many memories. (In fact, I used to be a moderator on their forums way back when as "RadioRob".)
You can find all of the various IPS hosted offerings at:
https://invisioncommunity.com/buy
Since your forum is listed in your signature, I went ahead and looked at it. Your community does not look to be too big and looks like it would do just fine on the Starter plan.
When you click the "Buy Starter" button, it will ask you to create your community. You'll choose "I want to use my own domain or subdomain". Enter your main domain www.lynxairways.com. From there, it will ask your billing information.
That should get your new account created. IPS will send you an email with details on how to get started transferring stuff. Once everything is copied over, one of the last things you'll do to point live users over is change the DNS from PowWeb to IPS by changing the name servers at Dotster.
Good luck!