Let F be a field of characteristic different from 2. It is shown that the problems of classifying

(i) local commutative associative algebras over F with zero cube radical,

(ii) Lie algebras over F with central commutator subalgebra of dimension 3, and

(iii) finite p-groups of exponent p with central commutator subgroup of order are hopeless since each of them contains

• the problem of classifying symmetric bilinear mappings UxU → V , or

• the problem of classifying skew-symmetric bilinear mappings UxU → V ,

in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.