177-Lutetium-PSMA Before Stereotactic Body Radiotherapy for the Treatment of Oligorecurrent Prostate Cancer, The LUNAR Study

Inclusion Criteria: * Oligorecurrent prostate cancer as determined by the presence of 1-5 asymptomatic lesions outside the prostate or prostate bed identified on PSMA PET/CT by local readers * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * No indication for urgent or emergent radiation * Histologic confirmation of prostate adenocarcinoma (histology from original treatment acceptable) * White blood cell count \>= 2.5 × 10\^9/L * Platelets \>= 100 × 10\^9/L * Hemoglobin \>= 9 g/dL * Total bilirubin =\< 1.5 × institutional upper limit of normal (ULN); or up to 3 × ULN if known history of Gilbert's syndrome * Alanine aminotransferase or aspartate aminotransferase =\< 3.0 × ULN or =\< 5.0 × ULN for patients with liver metastases * Serum creatinine =\< 1.5 × ULN or creatinine clearance \>= 50 mL/min * Serum albumin \> 3.0 g/dL * Partner and patient must use a method of birth control with adequate barrier protection, deemed acceptable by the principal investigator during the study and for 3 months after last study drug administration * Ability to understand, and willingness to sign, the written informed consent Exclusion Criteria: * Patients with neuroendocrine or small cell carcinoma of the prostate * Patients with castrate-resistant disease (i.e., PSA \> 0.5 ng/mL with serum testosterone \< 150 ng/dL) * Patients who received androgen deprivation therapy within 6 months of trial enrollment * Concurrent systemic therapy for a solid organ m…