Starting with generic stationary axially symmetric spacetimes depending on two spacelike isotropic orthogonal coordinates x1, x2, we build anisotropic fluids with and without heat flow but with wanishing viscosity. In the first part of the paper, after applying the transformation x1→J(x1)\documentclass[12pt]{minimal}
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\begin{document}$x^{1}\rightarrow J(x^{1})$\end{document}, x2→F(x2)\documentclass[12pt]{minimal}
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\begin{document}$ x^{2}\rightarrow F(x^{2})$\end{document} (with J(x1),F(x2)\documentclass[12pt]{minimal}
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\begin{document}$ J(x^{1}), F(x^{2})$\end{document} regular functions) to general metrics coefficients gab(x1,x2)→gab(J(x1),F(x2))\documentclass[12pt]{minimal}
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\begin{document}$ g_{ab}(x^{1},x^{2}) \rightarrow g_{ab}(J(x^{1}), F(x^{2}))$\end{document} with Gx1x2=0\documentclass[12pt]{minimal}
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\begin{document}$ G_{x^{1} x^{2}}=0$\end{document}, being Gab\documentclass[12pt]{minimal}
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\begin{document}$ G_{ab}$\end{document} the Einstein’s tensor, we obtain that G˜x1x2=0→Gx1x2(J(x1),F(x2))=0\documentclass[12pt]{minimal}
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\begin{document}$ \tilde{G}_{x^{1} x^{2}}=0\rightarrow G_{x^{1} x^{2}}(J(x^{1}),F(x^{2}))=0$\end{document}. Therefore, the transformed spacetime is endowed with an energy-momentum tensor Tab\documentclass[12pt]{minimal}
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\begin{document}$ T_{ab}$\end{document} with expression gabQi+\documentclass[12pt]{minimal}
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\begin{document}$ g_{ab}Q_{i}+$\end{document}heat term (where gab\documentclass[12pt]{minimal}
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\begin{document}$ g_{ab}$\end{document} is the metric and {Qi}\documentclass[12pt]{minimal}
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\begin{document}$ \{Q_{i}\}$\end{document}, i=1...4\documentclass[12pt]{minimal}
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\begin{document}$ i=1\ldots 4$\end{document} are functions depending on the physical parameters of the fluid), i.e. without viscosity and generally with a non-vanishing heat flow. We show that after introducing suitable coordinates, we can obtain interior solutions that can be matched to the Kerr one on spheroids or Cassinian ovals, providing the necessary mathematical machinery. In the second part of the paper we study the equation involving the heat flow and thus we generate anisotropic solutions with vanishing heat flow. In this frame, a class of asymptotically flat solutions with vanishing heat flow and viscosity can be obtained. Finally, some explicit solutions are presented with possible applications to a string with anisotropic source and a dark energy-like equation of state.
