HI. I live on an island where our water comes

The blue pipe with the elbow? Zip tie it to the overflow pipe, put a brick on it etc.

first, check all other taps in the house to see if water is flowing out of them properly. if not, check that your main stopcock is open. If it is, check if your neighbours have the same problem. If they have the same problem and your stopcock is open, check your cold water cistern. There's something called a ball valve in it that could be stuck causing the water to empty from the cistern. If this is what happened, sharply move the float arm to dislodge the valve and clean it.

Also check that any of your pipes aren't frozen, if you live in a particularly cold country.

I assume you have a modern compact cistern that probably has an economy flush facility?

I have very little experience of these. There are two possibilities - firstly the water level could be too high due to a leaking inlet valve and the water leaking into the bowl is from the internal overflow.

A friend has recently had one of these fitted to economise on water but it soon developed a constant leak into the lavatory pan. I checked the water level which was ok and then she called out the guys that fitted it - twice! The first guy took it apart and could find nothing wrong so put it back again and when the leak had stopped went away. The next day the leak was worse and so another guy arrived and did the same but in an effort to justify his existence replaced one small fibre washer and one small O ring even though they appeared to be ok.

The leak stopped and to date has remained stopped. Clearly that type of cistern needs to be assembled very carefully with the sort of precision not typical of plumbers or plumbing.

cisterns never completely empty as the setting of the mechanism to close the valve is activated when it gets heavier than the water supporting it
try allowing more water into the cistern on a fill by adjusting the water level control
get a plumber in to clean out the sewerage pipes as there appears to be an obstruction in the line

Pump probably needs to be serviced. It probably pressurizes a tank and when that tank reaches a pressure the pump is turned off. But just guessing.

Hi Juan
Unfortunately we can't see what you can see. The problem must be right in front of you but you are missing it. Eg Parts / Seals in backwards.
If you can't figure it out and it's that urgent then call a plumber.
Good Luck