Let h be a direct sum of n copies of a simple Lie algebra g. In 1994, Feigin, Frenkel, and Reshetikhin constructed a large commutative subalgbera of the enveloping algebra U(h). This subalgebra, which is an image of the Feigin—Frenkel centre, contains quadratic Gaudin Hamiltonians and therefore is known as a Gaudin subalgebra. By now it has been studied from various points of view and numerous generalisations have been obtained. We look at the `classical’ version of a Gaudin algebra, i.e., at its image in the symmetric algebra S(h). This image, say C, is Poisson-commutative and can be obtained from a suitable pair of compatible Poisson brackets on S(h) via the Lenard—Magri scheme. An advantage of the Lenard—Magri approach is a well-developed geometric machinery. For example, it allows us to show that C is algebraically closed in S(h). We will discuss also a generalisation to a non-reductive setting. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.