When I turn on the backlightning the LCD numbers disappear (the numbers are then fully transparent and quite not visible). Does anybody has an idea if it's possible to repair the instruments or should I throw it away?
You may be unaware but the contrast is adjustable on ST50s and this may be the problem, or if you had set it perhaps it has lost the setting. In order to access the adjustment you need to hold down 2 keys but look at the manual for which ones.
This may be an adjustment, but it does rather sound as though there's a high resistance somewhere that means the extra load of the backlighting means there isn't enough voltage to run the display. If the beast still works it's almost certainly within the display part - possibly a dry soldered joint.
If you can track down the dodgy joint using a voltmeter, it should be possible to resolder it.
LCD's age being exposed to sunlight. If this was a gradually occuring effect you may need to get the LCD changed. BTW which instrument are you using (depth, wind, speed?).
If it happned suddenly it could be the contrast settings or indeed a dodgy connection.