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If two differential operators Ln = n i=0 ui(x)∂ i x and Lm = m i=0 vi(x)∂ i x commute, then there is a non-zero polynomial R(z, w) such that R(Ln, Lm) = 0 (see [1]). The curve Γ defined by R(z, w) = 0 is called the spectral curve of the pair. If Lnψ = zψ and Lmψ = wψ, then (z, w) ∈ Γ. The dimension of the space of common eigenfunctions ψ is the same for almost all (z, w) ∈ Γ. The dimension of the space of common eigenfunctions of two commuting differential operators is called the rank of the pair. The rank is a common divisor of m and n. The coefficients of two commuting operators of rank 1 can be expressed explicitly in terms of Riemann theta-functions [2]. The case when the rank is greater than 1 is much more difficult. The first examples of commuting differential operators of non-trivial rank 2 and non-trivial genus g = 1 were constructed by Dixmier in [3] for the non-singular elliptic spectral curve w 2 = z 3 − α with an arbitrary non-zero constant α:where L, M is the commuting pair of Dixmier operators of rank 2 and genus 1. A general classification of pairs of commuting ordinary differential operators of rank greater than 1 was obtained by Krichever [4]. The general form of pairs of commuting operators of rank 2 for an arbitrary elliptic spectral curve was found by Krichever and Novikov [5]. The general form of pairs of operators of rank 3 for an arbitrary elliptic spectral curve (the general form of commuting operators of rank 3 and genus 1 is parametrized by two arbitrary functions) was found by Mokhov [6], [7]. Moreover, examples of commuting differential operators of genus 1 with polynomial coefficients have been constructed for any rank. However, even in those cases for which explicit formulae were obtained, the problem of determining commuting operators with polynomial coefficients is non-trivial and has not yet been completely solved. The problem of a complete description of commuting differential operators with polynomial coefficients was posed and discussed by Mokhov in [8]. In [9] examples were constructed of pairs of commuting operators L and M of rank 2 and arbitrary genus g:where the Ai are arbitrary constants, A3 ̸ = 0, and the ai are some constants. Examples of commuting differential operators of arbitrary genus and arbitrary rank with polynomial coefficients were constructed in [10]. In this paper we find new examples of pairs of commuting operators of rank 2. We have proved the following three theorems.

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We obtain topological restrictions on Maslov classes of monotone Lagrangian submanifolds of C n . We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in certain cases. Contents 1. Introduction 1 Acknowledgements 4 2. Preliminaries 4 2.1. Local Floer cohomology 4 2.2. Spectral sequences of the Floer algebra 5 3. Main theorems: the existence part 6 4. Main theorems: the restriction part 11 5. Some explicit examples of monotone submanifolds of C n 15 6. Monotone Lagrangian submanifolds of CP n 16 References 18

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