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We describe the Reidemeister spectrum $Spec_RG$ for $G = N_{rc},$ the freenilpotent 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 anygroup $N_{2c}$ for $c \geq 4$ satisfies to the $R_{\infty}$ property. As aconsequence we obtain that every free solvable group $S_{2t}$ of rank 2 andclass $t \geq 2$ (in particular the free metabelian group $M_2 = S_{22}$ ofrank 2) satisfies to the $R_{\infty}$ property. Moreover, we prove that anyfree solvable group $S_{rt}$ of rank $r \geq 2$ and class $t$ big enough alsosatisfies to the $R_{\infty}$ property.

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