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elimination of a real variable from a complex function

I am doing the stability analysis of the solutions of a complex integral
equation. In this context I am facing a problem which is detailed below.
Could some body help me resolve the problem.
Gautam C Sethia
The following complex integral equation is in a single real variable
\[Lambda] and a number of real parameters viz. A, \[CapitalOmega], v,
\[Alpha], k and n. The last two parameters i.e. k and n can have only
integer values. My objective is to eliminate \[Lambda] from this equation
and have an equation only in terms of the parameters. This should in
principle be possible as the complex equation can be reduced to a system of
two real equations and a single real variable \[Lambda] and that should
enable us to eliminate \[Lambda] from these two equations.

\[ImaginaryI]\[Lambda] + Integrate[((A*Cos[z] + 1)*Cos*(k*z + \[Alpha] +
      (1 - E^((-I)*(n + (\[Lambda]*Abs[z])/v))))/(2*Pi), {z, -Pi, Pi}] = 0

I tried the following:
1. I wrote the above equation as a set of two equations by separating into
real and imaginary parts. That is facilitated by the fact that variable
\[Lambda] as well as all other parameters are real.
2. In each of these two equations, the integration could be done separately
from -\[Pi] to 0 and 0 to \[Pi] so that we can get rid of the Abs function
on z.
3. The right hand side of these two equations can then be integrated in
mathematica (it takes about 10-15 minutes on my laptop) giving horrendous
looking analytical expressions.
4. I am at a loss at this point and have no clue how to eliminate \[Lambda]
from these two equations involving those two expressions.

Could you please suggest me a way to go about achieving my objective using
mathematica. Thanks...Gautam C Sethia

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