Hmm, the GraphicsComplex
apparently stores the information on all the surfaces, not just the lines we are interested about. We can extract the lines with Normal
:
plot = ContourPlot3D[Evaluate[eqs[[1]]],
{x, 3.75, 4.3}, {y, 1.25, 1.44}, {z, -.4, .4},
MeshFunctions -> Function @@ {{x, y, z},
eqs[[2, 1]]}, Mesh -> {{0}}, BoundaryStyle -> None,
ContourStyle -> None, AxesLabel -> Automatic];
lns = Cases[Cases[plot, _GraphicsComplex, All][[1]] // Normal,
_Line, All][[{1, 4, 5}]];
Graphics3D[lns, AxesLabel -> {x, y, z}, Axes -> True]
sols = Cases[lns, {_Real, _, _}, All];
eqs /. Equal -> List /. Thread[{x, y, z} -> sols[[1]]]
Row[{Graphics[lns /. {x_Real, y_, z_} :> {x, y},
Frame -> True, FrameLabel -> {x, y}],
Graphics[lns /. {x_Real, y_, z_} :> {x, z},
Frame -> True, FrameLabel -> {x, z}],
Graphics[lns /. {x_Real, y_, z_} :> {y, z},
Frame -> True, FrameLabel -> {y, z}]}]
The points along the lines are not very accurate solutions, but you can use them as starting points for iterative methods.