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In this paper we investigate the balanced condition (in the sense ofDonaldson) and the existence of an Englis expansion for the LeBrun's metrics on$C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balancedunless it is the flat metric. The second one shows that an Englis expansion ofthe Rawnsley's function associated to a LeBrun's metric always exists, whilethe coefficient $a_3$ of the expansion vanishes if and only if the LeBrun'smetric is indeed the flat one.

Some remarks on the Kaehler geometry of LeBrun's Ricci flat metrics on C^2