Question: Possibly incomplete solution set for a fixed-point equation

I am trying to solve a fixed-point equation.

K := Matrix(2, 2, {(1, 1) = 2, (1, 2) = 1, (2, 1) = 1, (2, 2) = 3})

q := 4

solve({a = (1-exp(-.5*K[1, 1]*a-.5*K[1, 2]*b))/(1+(q-1)*exp(-.5*K[1, 1]*a-.5*K[1, 2]*b)), b = (1-exp(-.5*K[2, 1]*a-.5*K[2, 2]*b))/(1+(q-1)*exp(-.5*K[2, 1]*a-.5*K[2, 2]*b))}, [a, b]);

However, no solutions are returned, and I get the warning message "Warning, solutions may have been lost."  How can I be sure that the full set of solutions has been returned?  (I should also say that, based on other cases of the same problem, I expect that there are two or three solutions.)

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