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We show that the potential wells with a central spike  V(x) = x^{10} + a x^8 + b x^6+ c x^4 + d x^2 + f/x^2 possess arbitrary finitemultiplets of elementary exact bound states. The strong asymptotic growth ofV(x) implies an ambiguity in the PT-symmetric quantization via complex boundaryconditions but the three eligible recipes coincide at our exceptionalsolutions.Comment: 15 pages and two figure

Spiked and 𝒫𝒯-symmetrized decadic potentials supporting elementaryN-plets of bound states