Relativistically invariant equations (βμ∂μ+ϱ)Ψ(x)=0 with singular ϱ, and detβ0 ≠ 0

We study relativistically invariant equations (β^μ?_μ+?)Ψ(x)=0 with a singular ?. The absence of a simple relation between these equations and their “conjugate” $\text{((β}{}^{\mathrm{\mu }}\text{)}{}^1\text{?}{}_{\mathrm{\mu }}\text{? ?}{}^1\text{)?(x) = 0}$ is pointed out. Assuming that Ψ(x) transforms under an arbitrary finite-dimensional representation of ISL(2,C), the problem of existence is investigated. It is shown that the assumption detβ⁰ ≠ 0 implies: all β^μ are invertible and the representation of ISL(2,C) is even- dimensional. The difficulties in the general treatment of the problem are discussed. A particular class of equations is illustrated by several examples.