First, to be able to create an
Application Page in SharePoint, you should create a SharePoint Solution then add an application page that must be deployed as farm solution only.
When you build your solution, it's packaged in a SharePoint Solution package (WSP) file. and you can use PowerShell
Install-SPSolution to deploy it to your farm.
After deploying your solution the Application page will be located in 14 hive at
C:\Program Files\Common Files\microsoft shared\Web Server Extensions\14\TEMPLATE\LAYOUTS\yoursolution name
And the WSP solution should be listed in
Central administration > System Management > Manage Farm Solutions
So you need to backup and restore a WSP solution rather than page as the following:
Backup WSP solution,
Now you will be able to backup all the WSP solution that deployed to your farm via PowerShell as the following:
$FolderPath = "c:\WSPSolutions"
foreach ($solution in Get-SPSolution)
$id = $Solution.SolutionID
$title = $Solution.Name
$filename = $Solution.SolutionFile.Name
Or you can just backup specific solution as
$file = (Get-SPFarm).Solutions.Item("AppPageSolution.wsp").SolutionFile
Restore WSP solution,
For more details check Install and deploy a SharePoint (WSP) Solution Package using SharePoint Shell