Answer by Michael E2 for Finding minimum and maximum of implicit functions

C0 = 3.201;FindMaximum[{z, 2*Ωxz == C0}, {{x, 1}, {z, 0.1}}](*  {0.0984138, {x -> 0.989292, z -> 0.0984138}}  *)C0 = 3.012;FindMinimum[{z, 2*Ωxz == C0}, {{x, 1}, {z, 0.1}}](*  {0.234294, {x -> 0.947234, z -> 0.234294}}  *)

To get the solution in Finding maximum or minimum of implicit functions to work, bound the variables away from the singularity (vertical tangent) at z == 0:

C0 = 3.201;Maximize[{z, 2*Ωxz == C0 && xL1 <= x <= xL2 && 0.01 <= z <= 0.2}, {x, z}](*  {0.0984138, {x -> 0.989293, z -> 0.0984138}}  *)C0 = 3.012;Minimize[{z, 2*Ωxz == C0 && xL1 <= x <= xL2 && 0.01 <= z <= 0.5}, {x, z}](*  {0.234293, {x -> 0.947233, z -> 0.234293}}  *)