DecomposingPolynomialSystems.jl is a Julia package that computes the symmetries that fix the parameters (specifically, the group of deck transformations) of a parametric polynomial system with finitely many solutions for generic parameters with a view towards decomposing the given polynomial system.

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using DecomposingPolynomialSystems
@var x[1:2] p[1:2]
F = System([x[1]^2 - x[2]^2 - p[1], 2*x[1]*x[2] - p[2]]; variables=x, parameters=p)
symmetries_fixing_parameters(F; degree_bound=1, param_dep=false)

The result of the last command is the object of type DeckTransformationGroup that contains 4 deck transformations acting on the unknowns x₁, x₂ of the polynomial system F:

DeckTransformationGroup of order 4
 structure: C2 x C2
 action:
  1st map:
   x₁ ↦ x₁
   x₂ ↦ x₂
  2nd map:
   x₁ ↦ (0.0 + 1.0*im)*x₂
   x₂ ↦ (0.0 - 1.0*im)*x₁
  3rd map:
   x₁ ↦ (0.0 - 1.0*im)*x₂
   x₂ ↦ (0.0 + 1.0*im)*x₁
  4th map:
   x₁ ↦ (-1.0 + 0.0*im)*x₁
   x₂ ↦ (-1.0 + 0.0*im)*x₂

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