On the Reidemeister spectrum and the $R_{\mathrm{\infty }}$ property for some free nilpotent groups

We describe the Reidemeister spectrum $Spe{c}_{R}G$ for $G=N_{rc},$ the free nilpotent group of rank $r$ and class $c,$ in the cases: $r\in \mathbb{N}$ and $c=1;$ $r=2,3$ and $c=2;$ $r=2$ and $c=3,$ and prove that any group $N_{2c}$ for $c\ge 4$ satisfies to the $R_{\mathrm{\infty }}$ property. As a consequence we obtain that every free solvable group $S_{2t}$ of rank 2 and class $t\ge 2$ (in particular the free metabelian group $M_2=S_{22}$ of rank 2) satisfies to the $R_{\mathrm{\infty }}$ property. Moreover, we prove that any free solvable group $S_{rt}$ of rank $r\ge 2$ and class $t$ big enough also satisfies to the $R_{\mathrm{\infty }}$ property.