The paper is devoted to the classical problem of inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus \(2\). Basic solutions \(F\) and \(G\) of this problem are obtained from a single-valued \(4\)-periodic meromorphic function on the abelian covering \(W\) of the universal hyperelliptic curve of genus \(2\). Here \(W\) is the nonsingular analytic curve \(W=\{u=(u_1,u_3)\in\mathbb{C}^2 : \sigma(u)=0\}\), where \(\sigma(u)\) is the two-dimensional sigma function. The gradient \(\nabla\sigma(u)=(\sigma_1(u), \sigma_3(u))\), where \(\sigma_i(u)=\frac{\partial\sigma(u)}{\sigma u_i}\), does not vanish at any point of the curve \(W\) (therefore \(W\) does not have singular points). The authors show that \(G(z)=F(\xi(z))\), where \(z\) is a local coordinate in a neighborhood of a point of the smooth curve \(W\) and \(\xi(z)\) is the smooth function in this neighborhood given by the equation \(\sigma(u_1, \xi(u_1))=0\). They obtain differential equations for the functions \(F(z)\), \(G(z)\), and \(\xi(z)\), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function \(G(z)\) into the Weierstrass elliptic function \(\wp\) under a deformation of a curve of genus \(2\) into an elliptic curve. This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with the sigma function. Section 3 tackles the field of meromorphic functions on the sigma divisor. Sections 4, 5 and 6 deals with some ultraelliptic integrals and relationship with a curve of genus \(1\). Section 7 contains an application of the addition theorem. For the two-dimensional sigma function, the addition theorem holds.