Irregularizability of the Dirichlet problem for one biharmonic system in $\mathbb{R}^4$

A linear homogeneous system of p first order differential equations in $\mathbb{R}^d$ is called biharmonic if each component of its arbitrary continuously differentiable solution satisfies the equation $\Delta^2u=0$, where $\Delta$ is the Laplace operator in $\mathbb{R}^d$. In this article we give an example of a biharmonic system in $\mathbb{R}^4$, which is neither a four-dimensional analogue of the Cauchy – Riemann system nor an elliptic pseudo-symmetric system. For this system we consider the Dirichlet problem in an arbitrary bounded region with a sufficiently smooth boundary. It is proved that at some point of the boundary the rank of the Lopatinski matrix of the Dirichlet problem is not maximal. It is also shown that at this point the limit problem for the considered Dirichlet problem is not uniquely solvable.