Good afternoon, excuse me for replying to this message after so long since you posted it, perhaps what I am going to comment on you have already found out...
These bikes are Spanish, manufactured by MOTOCREACIONES POUS S.A., also manufacturer of components for other bicycles and motorbikes (mainly Spanish) under the brand IDEAL.
The two white ones are mine, I also have them in red and green. I have seen them in yellow and blue.
As you say, they are with both front and rear "suspension" and there is also a version called STAR with telescopic front suspension which is more difficult to find. I leave you a link to my instagram where you can see the INTER version in green.
LOOPING STAR
At the rear and by means of the black knobs you can loosen and adjust the height of the bike a little.
The same brand also produced a pure BMX model and a Trialsin (biketrial) model.
Best regards.
P.D. Me acabo de dar cuenta de que eres español jajajaja Un saludo.