Decay spectroscopy of the odd-proton nuclei $^{249}$Md and $^{251}$Md has been performed. High-$K$ isomeric states were identified for the first time in these two nuclei through their electromagnetic decay. An isomeric state with a half-life of $2.4(3)$ ms and an excitation energy $geq 910$ keV was found in $^{249}$Md. In $^{251}$Md, an isomeric state with a half-life of $1.37(6)$ s and an excitation energy $geq 844$ keV was found. Similarly to the neighbouring $^{255}$Lr, these two isomeric states are interpreted as 3 quasi-particle high-$K$ states and compared to new theoretical calculations. Excited nuclear configurations were calculated within two scenarios: via blocking nuclear states located in proximity to the Fermi surface or/and using the quasiparticle Bardeen-Cooper-Schrieffer method. Relevant states were selected on the basis of the microscopic-macroscopic model with a deformed Woods-Saxon potential. The most probable candidates for the configurations of $K$-isomeric states in Md nuclei are proposed.
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