I tried using

     solve(ecT,[x], UseAssumptions) assuming x::imaginary;

at the end of the complex-roots-T.mw file.  All that was returned was [].

I think there is a subtle problem with how the problem is posed.  I don't see what it is, but it will probably be obvious when the solution surfaces :-)

I tried substituting s:= r*exp(I*theta) and the solutions still don't satisfy T.  I'm not sure what to do now....

Perhaps

I am trying to find Lyapunov functions for dynamic systems.  The
reference I am using for the subject is "Nonlinear Systems Analysis"
by Vidyasagar (second edition).  My reference has the following
definitions (p.147):

     A function V:R+ xR^n -> R is said to be a locally positive
definite function if (i) it is continuous (ii) V(t, 0)=0 for all t,
and (iii) there exist a constant r>0 and a function alpha of class
K
     (alpha(0) = 0, monotonic, alpha(r) -> infty as r -> infy) such
that

     (1)     alpha(||x||)<=V(t, x), for all t>=0 and for all x in some
region

     V is a positive definite function if (1) hold for all x in R^n

-Joel