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<p>The pseudopotential method is used to derive electron hole structures in a suprathermal plasma with a regularized <span class="inline-formula"><i>κ</i></span> probability distribution function background. The regularized character allows the exploration of small <span class="inline-formula"><i>κ</i></span> values beyond the standard suprathermal case for which <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M3" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi mathvariant="italic">κ</mi><mo>&gt;</mo><mn mathvariant="normal">3</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="40pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="29f515149cf2cfd4ee44932f3a6cedee"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="npg-30-277-2023-ie00001.svg" width="40pt" height="14pt" src="npg-30-277-2023-ie00001.png"/></svg:svg></span></span> is a necessary condition. We found the nonlinear dispersion relation yielding the amplitude of the electrostatic potential in terms of the remaining parameters, in particular the drift velocity, the wavenumber and the spectral index. Periodic, solitary wave, drifting and non-drifting solutions have been identified. In the linear limit, the dispersion relation yields generalized Langmuir and electron acoustic plasma modes. Standard electron hole structures are regained in the <span class="inline-formula"><i>κ</i>≫1</span> limit.</p>