You have to create the events You want for participating, also in ADMIDIO.
When You have created the event, You can "clip" the URL for the specific event, and this URL You can put in to the event in LuxCal.
You can put it in as a normal URL and put [Participate] just after the URL so Participate will be the text You see, and the libk will direct to the event in ADMIDIO.
A more sophisticated way, is to use the display1 to present the events for Your users.
You could then in extrafield1 or extrafield2 put an image and, in the same field You put the URL for the event in ADMIDIO.
Then - when a user clicks the image, he is directed to the ADMIDIO-event ( the same as in LuxCal ) and when he is signed in, he can participate the event. If he was already signed in, then he is directly in the event, and all he has to do is click on the "participation" button.
What is described here, both systems ( LuxCal and ADMIDIO ) can handle already.
Later hopefully, ADMIDIO will put on a milestone, that You can participate an event, even when You are not yet "accepted" as a user in ADMIDIO - You are only a "pending" user.
This will do, that You can have "public" events, that users can participate.
The user has of course to put in his name and email. The ADMIDIO-system will then send an email to him, and he then has two hours to ( in the mail there is a link ) accept/verify his participation. He can not log in to the system, before the ADMIDIO-admin accepts his membership, but his participation is registered/reserved on the event. Then when he is an accepted member, he can edit his participation.