I am using Silvios Xmailer to send mails. The mails are sent and the receiver gets them but there is no send item entry on the mail server for this mails, does anyone knows why?
Because the SMTP protocol has no concept of such things. If the SMTP server does not automatically save a copy of sent emails in the sender's folder, Xmailer would have to manually connect to the provider's IMAP server and place a copy of sent emails in the appropriate folder. XMailer is just a simple SMTP implementation, so it would not do that extra IMAP step. A full featured email program normally would.
When I send email with Thunbird or Macos-Mailprogramm then I can see the mails on the server in sent box
I can't speak for MacOS, but Thunderbird has a configuration that tells it which remote folder to place sent emails in, and it is usually smart enough to figure out what those folders are by default (but you can also change the configuration if desired).
but only the mails sent by xmail hav not send entry.
Then XMailer is not saving a copy of sent emails via IMAP, which makes sense for an SMTP-only implementation.