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We show that some Emden–Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the typed  2   z   /  d    χ  2    =   A   χ    −  λ   −  2z  n${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=A{\chi^{-\lambda-2}}{z^{n}}$    forλ  =    (   n  −  1   )   /  2$\lambda=(n-1)/2$    (class 1), andλ  =   n  +  1$\lambda=n+1$    (class 2). We find their corresponding invariants which reduce them to first-order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator the formd  2   ν   /  d    t  2    +a  d  ν   /  d   t   +   b   (   ν  −   ν  n    )=  0    ${\mathrm{d}^{2}}\nu/\mathrm{d}{t^{2}}+a\mathrm{d}\nu/\mathrm{d}t+b(\nu-{\nu^{n}})=0$   , where the coefficientsa  ,  b    $a,b$    depend only onλ  ,  n    $\lambda,n$   . For both classes, we write closed-form solutions in parametric form. The illustrative examples from astrophysics and general relativity correspond to two  n  = 2 cases from class 1 and 2, and one  n  = 5 case from class 1, all of them yielding Weierstrass elliptic solutions. It is also noticed that when  n  = 2, the EF equations can be studied using the Painlevé reduction method, since they are a particular case of equations of the typed  2   z   /  d    χ  2    =   F   (  χ  )    z  2${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=F(\chi){z^{2}}$   , whereF   (  χ  )$F(\chi)$    is the Kustaanheimo-Qvist function.

Two Integrable Classes of Emden–Fowler Equations with Applications in Astrophysics and Cosmology