Needle just gets stuck in a particular spot of the record ! What is going on?

Never experienced anything like that. I just bought this Japanese record, and though there are many hairlines it plays almost perfectly except for this spot. If you look at it while it's happenning the stylus moves back and forth endlessly and the same phrase keeps repeating forever without any noticeable distortion. I cannot see anything suspicious under a very strong light. The record is very clean, I cleaned it twice on a machine with Audio Intelligent three step system.
So what the hell is this? Groove wall damage? This is a Japanese record from seventies, I have a number of them from that time. That vinyl is excellent.
It is called a skip.
You can usually loosen whatever is adhered to the groove wall with good magnification and a round toothpick.  Since you've isolated the spot where it's occurring, it should be fairly simple.
Small junk in the bottom of the groove.

must be one of those locked groove recordings.
I gets more interesting. I cleaned the record once again and played it. First time it played through just fine, then I repeated a few times and it refuses to do it. The stylus is reasonably clean.
Get a strong magnifying glass and a sewing needle. A little speck of unloosened dirt is most probably the culprit. Sometimes what we can't see with the naked eye, looks like a boulder to the tip of the diamond. Once you see it, just gently probe with the needle until you have it out. 
Couple drops of dawn dish soap, scrub gently, rinse with clean lukewarm water..  Or let the dawn sit for a couple minutes to loosen the gunk. 
Dish soap didn't help. Could it be a manufactrurer's defect? I had other kinds of skips when the needle goes for a ride but never that. Anyway, I don't have a strong magnifying glass, and with the one I do I don't see anything.
Try a penny on top of the headshell.

Or you could slightly increase the tracking force.

I purchased "Are You Experienced" special edition, audiophile vinyl, numbered edition, etc. about 20 years ago and never played it until this year.There is an impassible spot in one song; A bridge from one groove-side to the next. From now on, will play all new vinyl to look for defects. I can probably use the straight pin (suggested by Oregonpapa) so will wait and see...

The record is not really valuable, anyway. Penny on top? Each time I play this side? No way. Or changing the tracking force each time. To hell with it, I got two more copies.
If not tracking near maximum tracking force for your cart try upping that.   Also make sure anti skate is not set too high. 
The tracking force is set just a little higher than average, I had it near minimum before, but it sounds better this way. The anti-skate is less than medium. Overall this is best to my liking for the records I listen to.
No other record like this, I have a few badly scratched ones and the stylus just goes through, with terrible sound but it goes through. Also, some of the records were so dirty before cleaning that they appeared to have gone through centuries of I don't know what. They play okay.
Dweller, you noted 
I purchased "Are You Experienced" special edition, audiophile vinyl, numbered edition, etc. about 20 years ago and never played it until this year.There is an impassible spot in one song; A bridge from one groove-side to the next.
Made me laugh - I have clear vinyl RYKO copies of LIve at Winterland and Radio One from maybe 25 years ago or so.  One of them (can't remember which one now) has what sounds like the same fault.  Haven't played it on my current rig, so can't confirm if it's actually a bridge.  Have to pull them out and run a test.

I also recall that one spot of a tune off Genesis' Wind & the Wuthering has something that sounds a bit like a slow static discharge in the middle of an acoustic guitar passage on every single copy I ever purchased, import or domestic.  That one must be in the master tape, but I never got a digital copy to see if it was edited out.  Another point of detail to explore sometime.

That's why we do the audiophile thing, I guess - to find all the joys of music.

Happy listening!
effischer: Remember this the next time you want to buy a "new sealed" copy of something on vinyl. It may be a load of crap. Reminds me, I bought an audiophile "King Crimson" (first album) at T.H.E. Show here in 'Vegas years ago. Yep, surfaced totally "hashed out" -what a joke.
Speaking of new copies, I recently bought by mistake Miles's "Bitches Brew" Mo-Fi reissue. Well, first the vinyl looks like it's hundred years old, real bad, and second they screwed up the re-mastering by making everything sound nice and even, the power of Miles's trumpet is just not there. Other instruments sound sometimes more distinct than on the original 2eye record, but that's not the point. I am debating either to put it in the closet or throw it away, either would do because I am not going to listen to it.
Inna, as in all things audio, there is someone who will think your Bitches Brew is the best version ever made. Sell it at a fair grade and recoup some money and make someone else happy. Win wins are hard to come by these days.

Hello Inna.

Have you tried Scotch Magic Tape, the translucent stuff? I cured a similar problem by
1. locating the exact spot on the record
2. cutting off a 5 cm piece of tape, reversing one end back on itself for 1 cm to form a non-sticky tab or handle, and
3. planting the 3 cm sticky tape onto the offending spot, rubbing in strongly, and removing by grasping the tab.

That tab is critical! You need to prevent the full length of tape from permanently affixing to the record!

The minute particle should go with the tape, or maybe on the second try. I tried this after heroic ultra-sonic cleaning, and for the first time since 1959, I was able to hear that track without the skip.

You have not mentioned what kind of RCM you are using but may I suggest taking your brush and going the opposite direction with the cleaning fluid and the rinse.

If you have access to a Spin Clean record cleaner, this is one of those times that they work very well. I use an eye dropper to apply the company specified amount of fluid over the entire surface of both pads. By applying the fluid at the top and bottom of the pads, and the remainder through out the centre of the pads, the lead in portion of the record gets much cleaner. I then go 30 spins in each direction. 15 right, 15 left, 15 right, and 15 left.It's the reversal of the brushes that really gets the gunk out of the grooves.

I have a vacuum RCM, but I still use the Spin Clean first to knock the bark off'em.
Hello terry9,
That's an interesting idea, thank you, I should try it. I guess, I will have to clean the record after that because there will be glue residue from Scotch.
regafan, I use Okki Nokki machine with Audio Intelligent three step cleaning system and Disc Doctor brushes. The motor can revolve the platter in both directions, that was one of the reasons why I bought it. The first step is enzyme pre-cleaner. With this particular record I let it soak for at least 15 minutes before vacuuming off.
I also use the Audio Intelligent three step cleaning system and like it very much. If you are able to correct the situation please be sure to post how you fixed the problem.