Clear["Global`*"]
M0 = 1.0;
\[Lambda] = 0.5;
For[kyv = -1.4, kyv < 1.4, kyv = kyv + 0.1,
Print[NSolve[
M0^6 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \[Lambda]) -
2 \[CurlyEpsilon]^4 \[Lambda]^2 Sqrt[-M0^2 +
kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] -
2 kyv M0 \[CurlyEpsilon]^3 \[Lambda]^2 (Sqrt[-M0^2 +
kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) +
M0^4 (-2 \[CurlyEpsilon]^4 + 4 \[CurlyEpsilon]^2 \[Lambda]^2 +
3 kyv^2 (-\[CurlyEpsilon]^2 + \[Lambda]^2) + \
\[CurlyEpsilon]^2 Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] - \[Lambda]^2 Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) +
2 kyv M0^3 \[CurlyEpsilon] (Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])] (Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] -
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) - \[CurlyEpsilon]^2 (Sqrt[-M0^2 +
kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) + \[Lambda]^2 (Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]])) +
M0^2 \[CurlyEpsilon]^2 (-kyv^2 (\[CurlyEpsilon]^2 +
3 \[Lambda]^2) - (\[CurlyEpsilon]^2 -
3 \[Lambda]^2) (-\[CurlyEpsilon]^2 +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]])) == 0, {\[CurlyEpsilon]}]]]
M0 = 1.0;
\[Lambda] = 0.5;
For[kyv = -1.4, kyv < 1.4, kyv = kyv + 0.1,
Print[NSolve[
M0^6 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \[Lambda]) -
2 \[CurlyEpsilon]^4 \[Lambda]^2 Sqrt[-M0^2 +
kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] -
2 kyv M0 \[CurlyEpsilon]^3 \[Lambda]^2 (Sqrt[-M0^2 +
kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) +
M0^4 (-2 \[CurlyEpsilon]^4 + 4 \[CurlyEpsilon]^2 \[Lambda]^2 +
3 kyv^2 (-\[CurlyEpsilon]^2 + \[Lambda]^2) + \
\[CurlyEpsilon]^2 Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] - \[Lambda]^2 Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) +
2 kyv M0^3 \[CurlyEpsilon] (Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])] (Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] -
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) - \[CurlyEpsilon]^2 (Sqrt[-M0^2 +
kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]) + \[Lambda]^2 (Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]] +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]])) +
M0^2 \[CurlyEpsilon]^2 (-kyv^2 (\[CurlyEpsilon]^2 +
3 \[Lambda]^2) - (\[CurlyEpsilon]^2 -
3 \[Lambda]^2) (-\[CurlyEpsilon]^2 +
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 -
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]]
Sqrt[-M0^2 + kyv^2 - \[CurlyEpsilon]^2 +
2 Sqrt[\[CurlyEpsilon]^2 \[Lambda]^2 +
M0^2 (\[CurlyEpsilon] - \[Lambda]) (\[CurlyEpsilon] + \
\[Lambda])]])) == 0, {\[CurlyEpsilon]}]]]


