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The solution for $P$ to $$P - G_1G_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0$$ is $$P=(1 +\lambda )G_1 G_2 A^{-1},$$ as one can check by substitution into $$G_1G_2P^\top(PAP^\top)^{-1}P=G_1 G_2(G_1G_2A^{-1})^\top\bigl(G_1G_2(G_1G_2A^{-1})^\top\bigr)^{-1}G_1G_2A^{-1}=G_1G_2A^{-1}.$$