What a mistake I make that the animation does not work:

(* Surface *) r[u_, v_] := { u, u Cos[v], u Sin[v]};  (* u - curve on surface *) v0[a_] := a; uconr := r[u, v0[a]];  (* v - curve on surface *) u0[b_] := b; vconr := r[u0[b], v];  (* ANIMATION *)  (* Surface *) rV = ParametricPlot3D[r[u, v], {u, -2 Pi, 2 Pi}, {v, 0, 2 Pi}];  (* u - curve on surface *) uconrA :=    ParametricPlot3D[uconr, {u, -2 Pi, 2 Pi}, PlotStyle -> Magenta];  (* v - curve on surface *) vconrA := ParametricPlot3D[vconr, {v, 0, 2 Pi}, PlotStyle -> Cyan];  Animate[Show[{rV, uconrA, vconrA},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}},    BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"},    Axes -> True, AxesLabel -> {x, y, z}, Ticks -> Automatic,    AxesStyle -> {Red, Green, Blue},    PlotLabel ->     Style[Framed["u - curve and v - curve on the cone surface",       FrameStyle -> Red], Bold, 14, Darker[Blue, .6],      Background -> Lighter[LightYellow]],    BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"}], {a, 0,    2 Pi}, {b, -2 Pi, 2 Pi}, AnimationRunning -> False] 

Animation works if the code looks like this:

(* Surface *) r[u_, v_] := { u, u Cos[v], u Sin[v]};  (* u - curve on surface *) v0[a_] := a; uconr := r[u, v0[a]];  (* v - curve on surface *) u0[b_] := b; vconr := r[u0[b], v];  (* ANIMATION *)  (* Surface *) rV = ParametricPlot3D[r[u, v], {u, -2 Pi, 2 Pi}, {v, 0, 2 Pi}];  Animate[Show[{rV,     ParametricPlot3D[r[u, v0[a]], {u, -2 Pi, 2 Pi},      PlotStyle -> Magenta],     ParametricPlot3D[r[u0[b], v], {v, 0, 2 Pi}, PlotStyle -> Cyan]},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}},    BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"},    Axes -> True, AxesLabel -> {x, y, z}, Ticks -> Automatic,    AxesStyle -> {Red, Green, Blue},    PlotLabel ->     Style[Framed["u - curve and v - curve on the cone surface",       FrameStyle -> Red], Bold, 14, Darker[Blue, .6],      Background -> Lighter[LightYellow]],    BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"}], {a, 0,    2 Pi}, {b, -2 Pi, 2 Pi}, AnimationRunning -> False] 

However, I would like to understand why the first code does not work. Understanding this fact is very important to me because I think it has to do with defining uconr, vconr and uconrA, vconrA. And this is the point of the matter for me.