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Decay spectroscopy of the odd-proton nuclei $^{249}$Md and $^{251}$Md hasbeen performed. High-$K$ isomeric states were identified for the first time inthese two nuclei through the measurement of their electromagnetic decay. Anisomeric state with a half-life of $2.8(5)$ ms and an excitation energy $\geq910$ keV was found in $^{249}$Md. In $^{251}$Md, an isomeric state with ahalf-life of $1.4(3)$ s and an excitation energy $\geq 844$ keV was found.Similarly to the neighbouring $^{255}$Lr, these two isomeric states areinterpreted as 3 quasi-particle high-$K$ states and compared to new theoreticalcalculations. Excited nuclear configurations were calculated within twoscenarios: via blocking nuclear states located in proximity to the Fermisurface or/and using the quasiparticle Bardeen-Cooper-Schrieffer method.Relevant states were selected on the basis of the microscopic-macroscopic modelwith a deformed Woods-Saxon potential. The most probable candidates for theconfigurations of $K$-isomeric states in Md nuclei are proposed.

First observation of high-$K$ isomeric states in $^{249}$Md and $^{251}$Md