When moving from an old server to a new one, it is possible to export all your keys.
On the old server, open the console, select "SSL/TLS certificates", then click on the "pen" icon next to the private key you want to export in the "Private keys" table.
The displayed screen will show a dump of the key which looks as follows:
-----BEGIN PRIVATE KEY-----
MIIEvgIBADANBgkqhkiG9w0BAQEFAASCBKgwggSkAgEAAoIBAQCtDhZFtSZbMkrl
....
3yj3VAdQtGPi4GmaGBJ1K3Zf
-----END PRIVATE KEY-----
All that text (including the delimiting lines
-----BEGIN PRIVATE KEY-----
and
-----END PRIVATE KEY-----
) is the dump of the private key and it fully defines it.
On the second computer, open the console, select "SSL/TLS certificates", then click on "Add". Choose a name for the key and set "Action" to "Import". Fill the "Key Contents" field with the dump of the private key you got from the other computer.
Press "OK" and the old private key should now be declared on the new computer.
You can also use a similar way to export your certificates.