<p>Bureau proposed a classification of systems of quadratic differential equations in two variables which are free of movable critical points, which was recently revised by Guillot. We revisit the quadratic Bureau-Guillot systems with the first and second Painlevé transcendent in the coefficients. We explain their birational equivalence by using the geometric approach of Okamoto’s spaces of initial conditions and the method of iterative polynomial regularisation, solving the Painlevé equivalence problem for the Bureau-Guillot systems with non-rational meromorphic coefficients. We explicitly determine the Hamiltonian functions associated with the systems, both for the cases where the system is Hamiltonian with respect to the standard 2-form and those for which are Hamiltonian with respect to the pull back induced by certain changes of variables. We also find that one of the systems related to the second Painlevé equation can be transformed into a Hamiltonian system via the iterative polynomial regularisation.</p>