### A bifurcation result for Sturm-Liouville problems with a set-valued term.

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We present a description of isochronous centres of planar vector fields X by means of their groups of symmetries. More precisely, given a normalizer U of X (i.e., [X,U]= µ X, where µ is a scalar function), we provide a necessary and sufficient isochronicity condition based on µ. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators ([X,U]= 0). We put also special emphasis on the mechanical aspects of isochronicity;...

We study the integrability of two-dimensional autonomous systems in the plane of the form $=-y+{X}_{s}(x,y)$, $=x+{Y}_{s}(x,y)$, where Xs(x,y) and Ys(x,y) are homogeneous polynomials of degree s with s≥2. First, we give a method for finding polynomial particular solutions and next we characterize a class of integrable systems which have a null divergence factor given by a quadratic polynomial in the variable ${({x}^{2}+{y}^{2})}^{s/2-1}$ with coefficients being functions of tan−1(y/x).

In this paper property (A) of the linear delay differential equation $${L}_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0,$$ is to deduce from the oscillation of a set of the first order delay differential equations.