Recently I got a bit interested in the possibility of finding spectrum of few interesting class of lattice quantum mechanical hamiltonians like Richardson's pairing hamiltonian, 1D Hubbard hamiltonian, 1D Heisenberg spin chains.
In this context I have this question, is the following generalized Richardson-Hubbard model hamiltonian exactly solvable (analytical feasibility of finding the spectrum) or integrable :
$$\mathbb{\hat{H}}=\sum_{x}^{}\hat{\mathbf{\Psi}}_{}^{\dagger}(x)\mathbb{A}(x)\hat{\mathbf{\Psi}}_{}^{}(x)+\sum_{x,y}\hat{\mathbf{\Psi}}_{}^{\dagger}(x)\otimes\hat{\mathbf{\Psi}}_{}^{\dagger}(x)\mathbb{B}(x,y)\hat{\mathbf{\Psi}}_{}^{}(y)\otimes\hat{\mathbf{\Psi}}_{}^{}(y)$$
where
$$\Psi(x)=\begin{pmatrix} c_{1}^{}(x) & \cdots & c_{n}^{}(x) & c_{1}^{\dagger}(x) & \cdots & c_{n}^{\dagger}(x)\end{pmatrix}_{}^{T}$$
with $c_{k}^{\dagger}(x)/c_{k}^{}(x)$ being fermionic creation/annhilation operator for creating/annhilating fermion of flavor $k$ at the site $x$ of a $1D$ periodic lattice for example. Further $\mathbb{A}(x)$ and $\mathbb{B}(x,y)$ are complex $n \times n$ and $n_{}^{2} \times n_{}^{2}$ matrices respectively. Just to be more general, here restriction to hermitian nature of $\mathbb{\hat{H}}$ is not assumed (so that it is possible to generalize techniques to open quantum systems setup).
If as such $\mathbb{\hat{H}}$ is not exactly solvable/integrable, what further restrictions are necessary on $\mathbb{A}(x)$ and $\mathbb{B}(x,y)$ for this problem to be exactly solvable/integrable : like restricting $x,y$ in the second summation of the hamiltonian definition to nearest neighbour and/or restricting $\mathbb{A}(x)$ and $\mathbb{B}(x,y)$ to be homogeneous (independent of $x$ and $y$) and/or imposing restrictions on structure of $\mathbb{A}(x)$ and/or $\mathbb{B}(x,y)$ matrices and/or dimensionality of flavor space ($n$) and so on (excluding trivial limits like non-interacting case ($\mathbb{B}(x,y)=\mathbb{O}_{n_{}^{2}\times n_{}^{2}}^{}$ for all $x,y$ of lattice) and/or all lattice sites decoupled case ($\mathbb{B}(x,y)=\mathbb{B}\delta_{xy}^{}$ - note here that i am having small $n$ case in mind of order atleast 4 and not exceeding 6)). One particular case I am interested in is $\mathbb{A}(x)$ and $\mathbb{B}(x,y)$ are homogeneous and $n=4$.
Specifically i am looking for the amenability of Richardson's ansatz or coordinate/algebraic/functional Bethe ansatz methods (i am still trying to figure out basic elements of algebraic Bethe ansatz, references along this direction will be extremely helpful too) to the exact solvability/integrability of $\mathbb{\hat{H}}$.
