We construct new examples of singular-hyperbolic attracting sets toward a possible classification. The novelty is that they accumulate their singularities in a pathological way or have no Lorenz-like singularities. The pathology will be explained using matrices with entries in {0,1}. The construction involves Dehn surgery [Goodman, S., 1983, Dehn Surgery on Anosov Flows, Geometric dynamics (Rio de Janeiro, 1981), 300-307, Lecture Notes in Mathematics, Vol. 1007 (Berlin: Springer)], templates [Ghrist, R.W., Holmes, P.J. and Sullivan, M.C., 1997, Knots and Links in Three-Dimensional Flows, Lecture Notes in Mathematics, Vol. 1654 (Berlin: Springer-Verlag)] and geometric models. Our examples are related to [Afraimovich, V.S., Bykov, V.V. and Shilnikov, L.P., 1982, On attracting structurally unstable limit sets of Lorenz attractor type (Russian). Trudy Moskov. Mat. Obshch., 44, 150-212; Bonatti, C., Diaz, L. and Viana, M., 2005, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III (Berlin: Springer-Verlag); Morales, C. and Pacifico, M.J., 2001, Mixing attractors for 3-flows. Nonlinearity, 14, 359-378].
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