I just can't find the steady state of a simple three country model

This model was used in the paper (the attached paper: “OIL AND THE MACROECONOMY: A QUANTITATIVE STRUCTURAL ANALYSIS”) ,but I can’t solve it’s steady state following the method (the attached image) given in the paper’s appendix A.
And I think this Lagrangian equation is a bit strange, because it pursues the maximization of the sum of the utility of the three countries.
wp_2010:04_Oil and the Macroeconomy_A Quantitative Structural Analysis.pdf (987.6 KB)

Lemme ask though if you are familiar with solving a planner’s problem

I first obtained the first-order derivation of the control variables in the model, and got 20 first-order conditions, plus 8 constraints, so I got a total of 28 equations, and then I called Matlab’s vapsolve function, using 28 equations Solve the values of 28 variables, but in the end there is no solution.

I tried two solutions. One is represented by the above picture, which is also suggested by the original text; the other is like this:

In both cases, I cannot find the steady state.

Oh, I see, can you share the intertemporal Euler equations you derived for the three households? What does it say in steady-state?

You should try to provide the analytical autarky solution as starting values.