Solving Linear Tensor Equations II: Including Parity Odd Terms in $4$-dimensions

In this letter, focusing in $4$-dimensions, we extend our previous results of solving linear tensor equations. In particular, we consider a $30$ parameter linear tensor equation for the unknown tensor components $N_{\alpha \mu \nu }$ in terms of the known (sources) components $B_{\alpha \mu \nu }$. Our extension now consists in including also the parity even linear terms in $N_{\alpha \mu \nu }$ (and the associated traces) formed by contracting the latter with the $4$-dimensional Levi-Civita pseudo-tensor. Assuming a generic non-degeneracy condition and following a step by step procedure we show how one can solve explicitly for the unknown tensor field components $N_{\alpha \mu \nu }$ and consequently derive its unique and exact solution in terms of the components $B_{\alpha \mu \nu }$.