Mixed function space with dolfinx - how to implement an integral constraint

In domain without decomposition, I am getting difficulty to relate the lagrange multiplier.
In reference tutorial multiphenicsx/tutorials/08_singular_poisson/tutorial_create_nullspace.ipynb at main · multiphenics/multiphenicsx (github.com), only one test function \delta v is used.

However, in implementing Lagrange multiplier \lambda

we have two test functions \delta \lambda for real space and \delta v for vector space. Can you tell me with analytical expressions how I can relate the tutorial with implementing constraints using lagrange multiplier deined in peroidic homo tutorial?