High degree of nonlinearities have three main sources:

JaimovichRebelo preferences.

SGUtype transaction costs.

Labor search frictions with intensive and extensive margin (e.g. wages, labor, values for firms and households are involved with probabilities derived from matching function that are nonlinear).
For example taking into account 2) and 3), my resource constraint looks like this:
\begin{matrix}y_t=c_t\left(1+A\left(\frac{c_t}{LGF_t}\right)+B\left(\frac{LGF_t}{c_t}\right)2\sqrt{AB}\right)+I_t\left(1+A\left(\frac{I_t}{LGF_t}\right)+B\left(\frac{LGF_t}{I_t}\right)2\sqrt{AB}\right)+\\
\xi_{1}\left(\frac{u_t}{vac_{1,t}}\right)^{\varphi}+\xi_{2}\left(\frac{u_t}{vac_{2,t}}\right)^{\varphi}\end{matrix}
Where LGF_t = m_{1,t}^{\theta}+m_{2,t}^{\theta}, and both m are two types of money, u_t is unemployment, vac_{i,t} are vacancies, where there are two types of labor (i\in{1,2}), c_t consumption, I_t investment, and the rest parameters. It seems a little too complicated (and maybe it is), but I simplified as much as possible and these are the essential features that I want my model to feature, for example y_t is just a CobbDouglas that uses capital and a CESaggregator of the two types of labor.
Also for example, the two types of money together with the LGF_t function allow for closed form expressions between them and their respective FOC, the nonclosed form part would become from the showed resource constraint. Besides, for example investment has also closed form expressión depending only on \delta k_t as usual, but for consumption it’s more tricky since it appears in firms and HHs values, FOCs in the nonlinear fashion of JaimovichRebelo preferences.
Actually, I know that the model without transaction costs has a steady state, in fact initially this is the model I’m trying to find the ss by myself, since so far I failed to do so with my own model (which includes the TC). The strategy I’m following is find successfully a steady state for the simpler model (which has a ss), and then try to extrapolate that method to my bigger model. Then for keeping things in the main line that I’m working on, this resource constraint would be more informative (just removing TC):
y_t=c_t+I_t+\xi_{1}\left(\frac{u_t}{vac_{1,t}}\right)^{\varphi}+\xi_{2}\left(\frac{u_t}{vac_{2,t}}\right)^{\varphi}
Which is the one I’m currently struggling altogether with the 1) and 3) sources of nonlineatities.