  In the present work, we obtain the constants of motion for isoperimetric
variational problems with time delay. We consider a constrained optimization
problem where the Lagrangian function defining the functional depends on time
delayed arguments. We prove the isoperimetric Euler--Lagrange and
DuBois--Reymond type optimality conditions and, in order to investigate the
constants of motion for this problem, we obtain a nonsmooth extension of
Noether's symmetry theorem for isoperimetric variational problems with delayed
arguments.
