in its spectrum within the class of the BPS vortex Hessian operator ${\cal H}^+$. The new development is one of the main novel proposals in this paper. With this objective in mind, but looking at a larger class of operators containing ${\cal H}^+$, we consider a general second-order $D\times D$ matrix PDO of the form $${\cal H} = - \textbf{I} \, \Delta + \textbf{u}^2 + \textbf{U}(\vec{x}) + \vec{\textbf{Q}} (\vec{x}) \cdot \vec{\nabla} \label{generaloperator}$$ where $\textbf{I}$ is the $D\times D$ identity matrix, $\Delta= \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2}$ is the 2D Laplacian, $\textbf{u}= {\rm diag}\,\{u_1,\dots,u_D\}$ is a constant $D\times D$ diagonal matrix determined by the asymptotic behaviour, $\vert \vec{x}\vert\to \infty$, of ${\cal H}$ and $\textbf{U}(\vec{x}) = (U_{ab}(\vec{x}))_{D\times D}$ with $a,b=1,2, \dots, D$, is a $D\times D$-matrix potential well. Besides $\vec{\textbf{Q}} (\vec{x}) =(\textbf{Q}_1(\vec{x}),\textbf{Q}_2(\vec{x}))$ is a vector field of matrices such that the last term in (\[generaloperator\]) reads $$\vec{\textbf{Q}} (\vec{x}) \cdot \vec{\nabla} = \Big(\sum_{i=1}^2 [\textbf{Q}_i(\vec{x})]_{ab} \, \partial_i\Big)_{D\times D}$$ We assume that $$\lim_{\vert\vec{x}\vert\rightarrow +\infty} \textbf{U}(\vec{x}) = \textbf{0} \hspace{0.5cm},\hspace{0.5cm} \lim_{\vert\vec{x}\vert\rightarrow +\infty} \textbf{Q}(\vec{x}) = \textbf{0} \label{generaloperator0}$$ which implies that the operator (\[generaloperator\]) asymptotically behaves as the PDO ${\cal H}_0=- \textbf{I} \, \Delta+ \textbf{u}^2 $. It is direct to check that the second-order small vortex fluctuation operator is 