Question: solutions of a transcendental equation in the real line and complex plane

Hello,

Sorry if the question is too naïve: I'm new to Maple.

I want to find the solutions, both in the real line and in the complex plane, of the following transcendental equation:

x+s*exp(x * sigma)-mu-s=0

mu is positive (say mu=6) and I am interested in the case in which s is both positive and negative (say, s=1 and s=-1).
By plotting s*exp(x * sigma) vs. mu+s-x (x real), and also using fsolve, I can see that there are two solutions when s=1 (although fsolve only reports the positive one) and none when s=-1.

Now comes the difficult case: to find the roots in the complex plane. Using the complex option with fsolve, I am able to find one of the four leading solutions for s=-1 (the one with possitive real and imaginary part), and don't know how to obtain more.
Writing x=a+ib  I'm able to obtain two (real) conditions (involving the cosine and the sine of b) that have to be fulfilled, and can obtain some graphical intuition of what is going on. 
On the other hand, I know that an analytical solution to the problem is related to the LambertW function, implemented on Maple, and so I would like to use all this power to plot, in a certain range, the (complex) roots to this transcendental equation, e.g, in the form Im(root) vs. Real (root) with the roots indicated by dots. 

Can anybody help me?

Best regards and thanks!