Do the fields exist without electric charges? I read in an old book on electrodynamics by Pauli that 

theoretically there does not exist any need of charges to be there. Fields can even exist without the charges but still independent fields have not been observed still. EMWs are also produced from any accelerated charges.

I want to know that have physicists found fields without electric charges today? I also want to know that if there actually does not exist any field without directly or indirectly originated from charges then there is something missing in the Maxwell equations or it is not like that?
I think that until we can experimentally prove the existence of the fields without charges we cannot tell the Maxwell equations a complete theory of the electrodynamics because it allows the fields without charges but if there exist some nature of the electromagnetic entity that it requires charges for fields then it is not explained in these equations.
 A: A propagating EM wave is a field that needs no charges. Likewise, a propagating gravitational wave is a field that needs no masses.
However in both cases there is no way (classically) to create the wave without a charge/mass. Considering EM, the divergence of the field is zero unless there is a charge, or put another way since field lines can only begin and end on charges all field lines would have to be infinitely long or looped.
A: I don't know cases in which the e.m. field can be produced without charges, however I can tell you how to produce an e.m. field without electric charges. Take a neutral particle, meet it with its antiparticle, have them clash, and you'll get gamma rays. The latter are electromagnetic. 
Also, gamma emission from excited nuclei, has not much to do with the fact that the nuclei contain protons. The gamma radioactivity originates in eliminating the excess of energy by excited nuclei (where the nuclear field dominates). 
If we can say or not that gamma emission from nuclei is without charges, I am not sure. But, in the case of annihilation the charges for generating e.m. field are charges of another field. 
(A good question would be here, are a particle and antiparticle opposite charges of some field? For the gravitational field, they are just masses.)
Good luck !  
