paramL = {sm -> 1.5*24, sw -> 1.*24, c -> 0.65, halfLwts -> 10, f -> 0.1*24, v1 -> 0.2*24, v2 -> 0.01*24, kfis -> 10, kfus -> 0.01, sf -> 0.5, sd -> 1.1};
sys = {(sw*wt)/((-mt*v2 + wt*(f - v2))/(c*v1) + 1) - (kfis/(kfis + kfus*(mt*sf + wt)))*wt*(Log[2]/halfLwts), (sm*mt)/((-mt*v2 + wt*(f - v2))/(c*v1) + 1) -  (kfis/(kfis + kfus*sf*(mt + wt)))*mt*sd*(Log[2]/halfLwts)} /. paramL;
vars = {wt, mt};
equilibria = Solve[sys == {0, 0}, vars, Reals];
saddles = Pick[equilibria, Sign[Det[D[sys, {vars}]]] /. equilibria, -1]

sepICS[p0_, eps_] := With[{p1 = p0 + eps*Norm[p0]*{Cos[t],  Sin[t]}}, p1 /. NSolve[Det[{p1 - p0, sys /. Thread[vars -> p1]}] == 0 && 0<= t < 2*Pi]];

sepICS[{997.6476, 0}, 10^-7]
{{997.648, 0.0000814314}, {997.648, 1.22177*10^-20}, {997.648, -0.0000837899}}

sepICS[{997.648, 0}, 10^-7]
{{997.648, 1.22177*10^-20}}