Visualization of the solution in the form of trajectories on the sphere
eq = {x[t] (-x[t] + x[t]^3 - x[t] y[t]^2 + y[t]^3 - y[t] z[t]^2 +
z[t]^3),
y[t] (x[t] + x[t]^3 - y[t] - x[t] y[t]^2 + y[t]^3 - y[t] z[t]^2 +
z[t]^3),
z[t] (x[t]^3 + y[t] - x[t] y[t]^2 + y[t]^3 - z[t] - y[t] z[t]^2 +
z[t]^3)};
sol = ParametricNDSolveValue[{eq == {x'[t], y'[t], z'[t]},
x[0] == Cos[b] Sin[c], y[0] == Sin[b] Sin[c],
z[0] == Cos[c]}, {x[t], y[t], z[t]}, {t, 0, 20}, {c, b}]
a = 1/Sqrt[14.]; Show[
Graphics3D[{{Green, Ball[]}, {Orange, PointSize[.05],
Point[{a, 2 a, 3 a}]}}, Boxed -> False],
ParametricPlot3D[
Evaluate[Table[sol[Pi/12, b], {b, 0, 2 Pi, .1}]], {t, 0, 10},
PlotRange -> All],
ParametricPlot3D[
Evaluate[Table[sol[Pi/3, b], {b, 0, 2 Pi, .1}]], {t, 0, 10},
PlotRange -> All]]
