Exploring highly deformed ground states involving the second intruder orbit in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>50</mml:mn> </mml:mrow> </mml:math> even-even nuclei

We present a systematic survey of even-even nuclei with $Z&gt;50$ to identify where a very large prolate configuration driven by the second intruder orbit emerges. Within the energy density functional theory framework, we find in representative cases a pronounced prolate minimum at quadrupole deformation ${\ensuremath{\beta}}_{2}\ensuremath{\approx}$ 0.3--0.4. A characteristic feature of these minima is a local enhancement of the hexadecapole (${\ensuremath{\beta}}_{4}$) component relative to nearby deformations, which is a clear fingerprint of the ${\ensuremath{\beta}}_{2}\text{\ensuremath{-}}{\ensuremath{\beta}}_{4}$ coupling expected for the second intruder orbit. Representative comparisons among three Skyrme interactions show a similar appearance of the highly deformed minimum and a local enhancement of ${\ensuremath{\beta}}_{4}$ at the prolate minimum, indicating qualitative robustness with respect to the interaction. The resulting maps highlight specific heavy nuclei where highly deformed ground states are anticipated.