P2m[y_, a_] := Sum[p[0, n2] Power[y, n2], {n2, 0, a}];

Ker[x_, y_] := μ1 c1 Power[x, 2] y + μ2 c2 x Power[y, 2] - (λ1 + λ2 + μ1 c1 + μ2 c2) x y +λ2 x + λ1 y;

Δ1[y_] :=
 Power[μ2 c2 Power[y,2] - (λ1 + λ2 + μ1 c1 + μ2 c2) y +λ2, 2] - 4 μ1 c1 λ1 Power[y, 2];

Θ1[y_] :=
 ArcTan[Sqrt[-Δ1[y]]/(μ2 c2 Power[y, 2] - (λ1 + λ2 + μ1 c1 + μ2 c2) y + 2 λ1 + λ2)];

δ1[θ_] :=
 Power[(λ1 + λ2 + μ1 c1 + μ2 c2 - 2 Sqrt[λ2 μ2 c2] Cos[θ]), 2] -
 4 μ1 c1 λ1;

χ[θ_] := (λ1 + λ2 + μ1 c1 + μ2 c2- 2 Sqrt[λ2 μ2 c2] Cos[θ] - Sqrt[δ1[θ]])/(2 μ1 c1);

Results = {};
λ1table = {1.10, 1.50, 1.80};
λ2table = {10.5};
atable = {2, 3, 4, 5};

For[i1 = 1, i1 <= 1, i1++, {
    For[i2 = 1, i2 <= 1, i2++, {
        For[i3 = 1, i3 <= 1, i3++, {

            NumVal = {λ1 -> λ1table[[i1]], μ1 -> 1.0, c1 -> 1, λ2 -> λ2table[[i2]], μ2 -> 1.0, c2 -> 10, a -> atable[[i3]]};

            Case = λ1 > μ1 c1 && μ1 c1 + μ2 c2 <λ1 + λ2 /. NumVal;

            If[Not[Case], Goto[end]];

            {Ny1, Ny2} = Sort[NSolve[Δ1[y] == 0 /. NumVal, y], Less][[{1, 2}, 1, 2]];

            Iϕ1[y_, x_] := Iϕ1[y, x] = (x (λ2 - μ2 c2 Power[y, 2]) Θ1[y])/(y Ker[x, y]) /. NumVal;

            Nϕ1[x_?NumericQ] := Nϕ1[x] = Exp[(1/π) NIntegrate[Iϕ1[y, x], {y, Ny1, Ny2}, Method -> {Automatic, "SymbolicProcessing" -> 0} ]];

            IΦ1[ξ_, y_] := IΦ1[ξ, y] = (y (μ2 c2 Power[ξ, 2] - λ2) Θ1[ξ])/(ξ (ξ - y) (μ2 c2 y ξ - λ2)) /. NumVal;

            NΦ1[y_?NumericQ] := NΦ1[ y] = (1/π) Re[ NIntegrate[IΦ1[ξ, y], {ξ, Ny1, y, Ny2}, Method -> {"PrincipalValue", "SymbolicProcessing" -> 0} ]];

            NSinguy[θ_] := NSinguy[θ] = (λ2 Sec[θ])/(2 Sqrt[ c2 λ2 μ2]) /. NumVal;

            Iβ[θ_, n_] := Iβ[θ, n] = (Nϕ1[ χ[θ] ] χ[θ] Sin[(n+ 1) θ] Sin[θ])/(λ1 + λ2 - (μ1 c1 +μ2 c2) χ[θ] ) /. NumVal;

            β[ n_?NumericQ] := ((2 λ2 μ2 c2)/(λ1 π))Re[NIntegrate[Iβ[θ, n], {θ, 0, π}, Method -> {Automatic, "SymbolicProcessing" -> 0}]] /. NumVal;

            Iα1Aux[θ_, y_, k_] := Iα1Aux[θ, y, k] = ((λ2 - μ2 c2 Power[y, 2]) Power[y, k - 1] Sin[Θ1[ y]] Exp[-NΦ1[y]])/(λ2 - 2 y Sqrt[λ2 μ2 c2] Cos [θ]) /. NumVal;

            α1Aux[n_?NumericQ, k_?NumericQ, θ_?NumericQ] := α1Aux[n, k, θ] = If[ Reduce[Ny1 < NSinguy[θ] < Ny2 && θ < (π/2)], NIntegrate[ Iα1Aux[θ, y, k], {y, Ny1, NSinguy[θ], Ny2}, Method -> {"PrincipalValue", "SymbolicProcessing" -> 0}], NIntegrate[Iα1Aux[θ, y, k], {y, Ny1, Ny2}, Method -> {Automatic, "SymbolicProcessing" -> 0}]];

            I1α1[θ_, n_] := I1α1[θ, n] = (Nϕ1[ χ[θ] ] Sin[(n + 1) θ] Sin[θ])/(λ1 + λ2- (μ1 c1 + μ2 c2) χ[θ] ) /. NumVal;

            α1[n_?NumericQ, k_?NumericQ] := ((2 μ2 c2)/Power[π, 2]) NIntegrate[ I1α1[θ, n]*α1Aux[n, k, θ], {θ, 0, π}, Method -> {Automatic, "SymbolicProcessing" -> 0}] /. NumVal;

            Iα2[θ_, n_, k_] := Iα2[θ, n, k] = χ[θ] Sin[(n + 1) θ] ((μ1 c1 χ[θ] - λ1- λ2) Sin[(k - 1) θ] + Sqrt[λ2 μ2 c2] Sin[ k θ])/((μ1 c1 + μ2c2) χ[θ] - λ1 - λ2) /. NumVal;

            α2[n_?NumericQ, k_?NumericQ] := α[n, k] = (2 Power[Sqrt[λ2/(μ2 c2)], k]/π) NIntegrate[ Iα2[θ, n, k], {θ, 0, π}, Method -> {Automatic, "SymbolicProcessing" -> 0}] /. NumVal;

            α[n_?NumericQ, k_?NumericQ] := Re[α1[n, k] + α2[n, k]];

            plist = Flatten[Solve[ Table[ Power[Sqrt[λ2/(μ2 c2)], n] p[0, n + 1] == β[n] p[0, 0] + Sum[α[n, k] p[0, k], {k, 0, a /. NumVal}] /. NumVal, {n, 0, a - 1 /. NumVal}], Table[p[0, n], {n, 1, a /. NumVal}]], 1];

            NP2mp0[y_] := NP2mp0[y] = P2m[y, atable[[i3]]] /. NumVal /. plist /. {p[0, 0] -> 1};

            IP1[x_, y_] := IP1[x, y] = ((λ2 - μ2 c2 Power[y, 2])/(y Ker[x, y]) ) NP2mp0[ y] Sin[Θ1[y]] Exp[-NΦ1[y]] /. NumVal;

            NP1[x_?NumericQ] := NP1[x] = Nϕ1[x] p[0, 0] (1 + (λ1/(π λ2)) x NIntegrate[ IP1[x, y], {y, Ny1, Ny2}, Method -> {Automatic, "SymbolicProcessing" -> 0}]) /. NumVal;

            𝒫2 = NP1[1];

            𝒫1 = NP2m[1] /. NumVal;

            solp00 = Solve[λ1 𝒫1 + λ2𝒫2 == λ1 + λ2 - μ1 c1 - μ2 c2/. NumVal, p[0, 0]];

            ℒ = {𝒫1, 𝒫2} /. solp00;

            AppendTo[Results, {λ1, μ1, c1, λ2, μ2, c2, a, ℒ} /. NumVal];

            Label[end];

            Clear[Iϕ1, Nϕ1, IΦ1, NΦ1, Iβ, β, Iα1Aux, α1Aux, I1α1, α1, Iα2, α2, IP1, NP2mp0];
        }]
    }]
}]

Results