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In this work we construct the eigenstates of the most general spin-1/2Richardson-Gaudin model integrable in an external magnetic field. This includesthe possibility for fully anisotropic XYZ coupling such that the $S^x_iS^x_j$,$S^y_iS^y_j$ and $S^z_iS^z_j$ terms all have distinct coupling strengths. Whileinsuring that integrability is maintained in the presence of an external fieldexcludes the elliptic XYZ model which is only integrable at zero field, thiswork still covers a wide class of fully anisotropic (XYZ) models associatedwith non skew-symmetric r-matrices.  The eigenstates, as constructed here, do not require any usable Bethe ansatzand therefore: no proper pseudo-vacuum, Bethe roots, or generalised spinraising (Gaudin) operators have to be defined. Indeed, the eigenstates aregenerically built only through the conserved charges which define the model ofinterest and the specification of the set of eigenvalues defining theparticular eigenstate. Since these eigenvalues are, in general, solutions to asimple set of quadratic equations, the proposed approach is simpler toimplement than any Bethe ansatz and, moreover, it remains completely identicalindependently of the symmetries of the model. Indeed, the construction removesany distinction between XYZ models and XXZ/XXX models and, generically, thatbetween models with or without U(1) so that any difficulties associated withthe use of a Bethe ansatz in any of these cases are avoided.

"Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models