...is that the problem of rotational motion must be solved with the help of GRT. The mistake made by Hafele and Keating was treating this problem using SRT which, on the other hand, required some false assumptions which led to this paradox. Eventually, Hafele and Keating proved the GRT while they were convinced that they proved the SRT.
CORRECT SOLUTION 
WRONG SOLUTION 
substituting we eventually obtain: 
Assuming temporarly an intertial motion we obtain factor describing the time dilation as a result of inertial motion: (1V^{2}/c^{2}) and introducing a gravitational field we obtain factor describing a gravitational time dilation: (1r_{s}/r) where r_{s} it's Schwarzschild radius of a gravitational source
and r is the radius, the objetct is rotating around 
For the first glance, the both formulas  the one derived on the base of GRT (the "correct" solution) and the one derived on the base of SRT (the "wrong" solution) are identical, therefore the way of derivation of the formula does not matter form point of view
of computational results. THEREFORE, IS THERE ANY REASON TO TALK ABOUT IT? YES  IF WE WANT TO FIND OUT INFORMATION CONCERNING THE ORIGIN OF THE TIME DILATION PHENOMENON 

In the case of proper  GRT based  solution, the Velocity in the formula (on the yellow background) is corresponding to the rotational motion, It means that the source of time dilation is the change of the velociny  more detail: the change of direction of the velocity 
In case of wrong  SRT based  solution the velocity in the formula is corresponding to the linear motion. It means that the source of time dilation is only velocity and change of the velocity has nothing to do with the time dilation phenomenon. 
Starting either from porper or wrong approach we obtain the same formula, then the result of computation does not depend on the choice of method, the problem is considered with. Computational results of the time dilation phenemoenon will be almost identical as well for the wrong as the correct method. The difference will appear when we try find the real mechanism responsible for the time dilation. Applying the wrong method
we are convinced that only the velocity causes the time dilation. Therefore the majority of physicists belives that the time
dilation in case of motion of GPS satelites or jets in the Hafele and Keating experimemt etc. are the proof for the SRT. The true mechanisms responsible for time dilation phenomenon are the velocity together with the change of the velocity an the problem can be solved properly with the help of GRT. The SRTtype formula is only the aproximation of the GRT solution and it can give us the false impression that the time dilation is a result of SRT while in fact it is result of GRT (although computations made with SRT method give correct results).
If the time dilation in case of the rotational motion can not be a result of SRT as it has been assumed up to now, then maybe other experiments which were assumed to be the proof of the SRT, were in fact the proof of GRT as it took place in case of rotational motion??? 