A cubature formula Q is an approximation of an n-dimensional integral I. Q is exact for the space spanned by the polynomials \(f_ 1,...,f_ d\) if it verifies the system of equations: \(Q[f_ i]=I[f_ i]\quad i=1,...,d.\) The unknowns are knots and weights of the cubature formula. We suppose that there are as many unknowns as equations. For searching solutions to this system, we construct a family of systems depending continuously on a parameter t: \(Q[f_ i(t)]=I[f_ i(t)]\quad i=1,...,d,\) coinciding with the previous system for \(t=1\) and whose solutions at \(t=0\) are easily computed. The solution curves originating from these solutions are followed numerically and may yield a solution for \(t=1\).